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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I AN INTRODUCTION - -^ TO ALGEBRA; WITH , NOTES AND OBSEBYATIOXS : DESIGNED For the Use of Schools and Places of Public £ducaUou. BY JOHJ^" BOjsrj\rYCASTLE, OF THE ROTAT. MILITAKY ACADKMY) WOOLWICH. Ing^nuas didicisse fideliter artes EmolUt mores> nee finit esse feros. Ovid. THE SECOND AMERICAN EDITION, EEVISJ^B ANP CORRECTED. FHIIiABEIiPHIA ; PXnUJSHED BT KIMBER AM> CONRAD, NO. 93, MARKET STREj^T.^ * T. d* G. Fahorrt printen* 1811. !'C 39335^ (l. s.) DISTRICT OP PENNSYLVANIA, to wit: Bs IT REMEMBERED, That on the eleventh day o bruaiy, in the thirty .fifth year of the Independer the United States of America, A. D. 1811, KiMBBR ANO Conrad, of the said cUstrict, have deposited in this office the title of a the right whereof they claim as proprietors, in the words foil o to wit : An Introduction to Algebra ; with Notes and Observations signed for the Use of Schools and Places of Public Educ By John Bonnycastle, of the Royal Military Academy, ^ wich. — Ingenuas didicisse fideliter artes Emollit mores, ncc finit ecse feros. Ovid. The second American edition, revised and corrected*. In conformity to the act of the congress of the United State tied, ** An act for the encouragement of learning, by secur copies of maps, chatts, and books, to the authors and pre oi such copies during the times therein mentioned." An» the act, entitled, ** An act supplementary to an act, entitle act for the encouragement of learning, by securing the r maps, charts, and books, to the authors and proprietors copies during the times therein mentioned," and exter benefits thereof to the arts of designing, engraving, and et torioal and other prints.** D. CALDWEI Clerk of the District of Pair THE acknowledged merits of the folloxving work J and its peculiar fitness for the use of ma- thematical students^ has induced the Editor to give it a careful revision: the errors of former editions and ^me improprieties in the original 'e corrected and removed. E. L. New-Garden Boarding-school, U mo. 1810. A 4 PREFACE. THE powers of the mind, like those of the body, are increased by frequent exertion ; application and industry ««ipply the place of genius and invention ; and even the creative faculty itself may be strengthened and improved by use and perseverance. Uncultivated nature is uni- formly rude and imbecile^ it being by imitation alone that we at first acquire knowledge, and the means x)f extending its 1x)unds. A just and perfect acquaintance with the simple elements of science, is a necesBary steg towards our future progress and advancement : and this>. assisted by laborious investigation and habitual inquiry, will constantly lead to eminence and perfection. Books of rudiments, therefore, concisely written, welF digested, and methodically arranged, are treasures of inestimable value; and too many attempts cannot be made to render them perfect and complete. When the fil*3t principles of any art or science are firmly fixed and rooted in- the mind, their application soon becomes a2 ▼i PREFACE. easy, pleasant, and obvious ; the understanding is delight- ed and enlarged ; we conceive clearly, reason distinctly, and form just and satisfactory conclusions. But^ on the contrary, when the mind, instead of reposing on the sta- bility of truth and received principles, is wandering in doubt and uncertainty, our ideas will necessarily be con- fused and obscure ; and every step we take must be at- tended with fresh difficulties and endless perplexity. Tjiat the grounds, or fundamental parts, of every science are dull and unentertaining, is a complaint uni' versally made, and a truth not be denied ;f but, then, what is obtained with difficulty is usually remembered with ease ; and what is purchased with pain is possessed \i4th pleasure. The seeds of knowledge are sown in every soil, but it is by proper culture alone that they are cherished and brought to maturity. A few years of early and assiduous application never fails to procure us the reward of our industry ; and who is there, who knows the pleasures and advantages which the sciences afford, that would think his time mispent, or his labours use« less ? Riches and honours are the gifts of fortune, ca- sually bestowed, or hereditarily received, and arj fre- quently abused by their possessors ; but flie superiority of wisdom and knowledge is a pre-eminence of merit which originates with the man, and is tlie noblest of all distinctions. Nature, bountiful and wise in all things, has provided us with an infinite variety of scenes, both for our instruc- tion and entertainment; and^ like a kind and indulgent PREFACE. vii parent} admits all her children to an equal participation of her blessing^. But^ as the modes, situations, and cir- cumstances of life are various, so accident, habit, and education have each their predominating^ infiuence, and give to every mind its particular bias. Where examples of excellence are wanting, the attempts to attain it are but feW'; but eminence excites attention, and produces imitation. To raise the curiosity, and to awaken the listless and dormant powers of younger minds, we have only to point out to them a valuable acquisition, and the means of obtaining it. The active principles are imme- diately put into motion, and the certainty of the conquest is ensured from a determination to conquer. B«t, of all the sciences which serve to call forth this spirit of enterprise and inquiry, there are none more eminently useful than the mathematics. By an early attachment to these elegant and sublime studies, we ac- quire a habit of reasoning, and an elevation of thought, which fixes the mind, and prepares it for every other pursuit. From a few simple axioms, and evident prin- ciples, we proceed gradually to the most general propo- sitions and remote analogies: deducing one truth f^m another, in a chain of arginnent well connected and lo- gically pursued; which brings hs, at last, in the most satisfactory manner, to the conclusion, and serves as a general direction in all our inquiries after truth. And it is not only in this respect that mathematical learning is so highly valuable; it is, likewise, equally estimable for its practical utility. Almost all the works • • • t»i PREFACE. of art) and devices of man, have a dependence up principleSf and are indebted to it for their origii perfection. The cultivation of these admirable sci is, therefore, a thing of the utmost importance ought to be considered as a principal part of every fal and v^ell-regulated plan of education. They ar guide of our youth, the perfection of our reason^ ar foundation of every great and noble undertaking. From these considerations I have been induc( undertake an introductory course of mathematical sci and, from the kind encouragement v^hich I have hit i*eceived, am not writhout hopes of a continuance c same candour and approbation. Considerable pn as a teacher, and a long attention to the difHcultie! obstructions which retard the progress of fearne general, have enabled me to accommodate mysel ipore easily to their capacities, and understandings. as an earnest desire of promoting and diffusing i: knowledge is the chief motive for this undertaking, i pains or attention shall be wanting to make It as com and perfect as possible. The subject of the present performance is A BRA^ which is one of the most important and u branches of those sciences, and may be justly consic as the key to all the rest. Geometry delights u the simplicity of its principles, and the elegance c demonstrations. Arithmetic is confined in its ob and partial in its application: but algebra, or analytic art, is general and comprehensive, and PREFACE. ix be appUed with success in all cases where truth is to be obtained and proper data can be established. To trace this science to its birth, and to point out the various alterations and improvements it has undergone in its progress, would far exceed the limits of a preface. It wiU be sufficient to observe, that the invention is of the highest antiquity, and has challenged the praise and admi- ration of all ag^s. Diofihantus appears to have been the first, among the ancients, who applied it to the solution of indeterminate or unlihiited problems ; but it is to the mo* demsthat we are principally indebted for the mosfcurious i^ement^ of the art, and its great and extensive useful- ness in every abstruse and difficult inquiry. JW w/on, Ma* clauririj Saunderson, Simfiaony and Emeraotij are those, of oor own countrymen, who have particullarly excelled in ^8 respect ; and it is to their works that I would refer the young student, as the patterns of elegance and perfection. The following compendium is formed entirely upon the iiiodel of those writers, and is intended as a useful and ne- cessary introduction to them. Ahnost every subject, ^hich belongs to pure algebra, is concisely and distinctly ^tedof; and no pains have been spared to make the ^hole as easy and intelligible as possible. A great num- ^r of elementary books have already been written upon ^s subject ; but there are none, which I have yet seen, but what appear to me to be extremely defective. Besides being totally unfit for the purpose of teaching, they are generally calculated to vitiate the taste, and mislead the X PREFACE. judgment A tedious and inelegant method prevails through the whole, so that the beauty of the science is generally destroyed by the clumsy and aukward manner in which it is treated ; and the learner, when he is after- wards introduced to some of our best writers, is obliged to unlearn and forget every thing which he has been at so much pains in acquiring. It is in the sciences as in every branch of polite litera- ture ; there is a certain taste and elegance which is only to be obtained from the best authors, and a judicious use of their instructions. To direct the student in his choice of books, and to prepare him properly for the advantages he may receive from them, is, therefore, the business of every writer who engages in the humble, but useful task of a preliminary tutor. This information I have been care- ful to give, in every part qf the present performance, where it appeared to be in the least necessary ; and, though the nature and confined limits of my plan admit- ted not of diffuse observations, or a formal enumeration of particulars, it is presumed nothing of real use and im- portance has been omitted. My principal object was to consult the ease, satisfaction, and accommodation of the learner ; and the favourable reception the work has met with from the public, has induced me to give this edition an attentive and careful revisal. THE CONTENTS. Page Definitions - - - - ^ . . 13 Ad^lion of algebra 16 Subtraction of algebra - - • - , 20 Multiplication of algebra - - - - <> 3 1 Division'of algebra ^ - • - - - 24 Algebraic fractioni^ - - - - - 28 Involution, or the raising of powers • - - 43 Evolution, or the extraction of roots - ^-v 47 Of irrational quantities^ or surds - - - 52 Of infinite series - - . - . - 63 Of arithmetical proportion - - - - 70 Of geometrical proportion ... " 73t Of simple equations - - - - - 76" Of quadratic equations 99 Of the nature and formation of equations in general 113 Of the resolution of equations by various methods 423 Of the resolution of cubic equations - - - 128 Of the resolution of biquadratic equations - ISl To find the roots of equations by approximation and converging series 1^6 To find the roots of pure powers in numbers - 1 41 To find the roots of exponential equations - - 14^ Of indeterminate or unlimited problems - - 145 Of diophantine problems - - - - 152 Of the summation and interpolation of infinite series 1 63 Of logarithms 190 A collection of miscellaneous questions - - 21^ ALGEBRA. DEFINITIONS- ALGEBRA is the art of computing by symbols. 1. L.ike quantitiea are those which consist of the same letters. 2. Unlike quantities are those which consist of diffe- rent letters. 3. Given quantitiea are those whose values are known. 4. Unknown quantities are those whose values are un- known- 5. Simfile quantities are those which consist of one term only. 6. Comfiound quantities are those which consist of se- veral terms. 7. Positive or affirmative quantities are those which are to be added. 8. Negative quantities are those which are to be sub- tracted. 9. Like signs are all affirmative (-f)j or all negative (-)• 10. Unlike signs are when some are affirmative (4-^ and others negative ( — ). 11. The co»efficient of any quantity is the number pre- fixed to it. B 14 EXPLANATION OF 12. A binomial quantity is one consisting of two terms; a trinomial of Uiree terms; a quadrinomial of four, &c. 13. A residual quantity is a binomial where one of the terms is negative. i4. llic flower of a quantity is its square, cube, bi- quadrate, &c. 15. The index or ex/ionent of a quantity is the number •which denotes its root or power. 16. A surd or irrational i/uaniity is that which has no exact root. 17. A rational quantity is that which has no radical sign (v/) or index annexed to it. .18. The recifirocal o[ dA^y quantity is that quantity in- verted, or unity divided by it. EXPLANATION OF THE CHARACTERS. + Is the sign of addition. — _ of subtraction. X . of muhiplicalion. -5- — of division. : : : : of proportion. y/ — . of the square root. ^ — of the cube root. s= — of equality. Thus, a 4" ^ sliows that the number represented by b is to be added to tliat represented by a. a— 6 shows that the number represented by b is to be subtracted from that, represented by a. aTjb reprtsents the difi'crence of a and b when it is not known which is the ij^reatest. ab^ OT axb^ or a.b denotes the product of the numbers represented by a and b. tml: characters. 15 a a-T-by or — , shows that the nunib's^r represented by a is lobe divided by that reprcsciUed by /;. a \ b I : c I d denotes iiiat a is u\ the same proportion to A as c is to d, jc = u — b + c is un e(iuutioii, siiowing that x is equal to the differ*, nee of a and b^ ud\)cd lo the quantity c, 1 _i V^a, or a*', is the square root of :; ; ^/a^ or a^, is the cube root of a; and u" is the mU root of r?. c2 is the square of a; a^ u.^ cube of a\ w^ the fourth power of a ; and a"* the //nil power of a, — is the reciprocal of — , and — the recipi*ocal of a. a a 4- 6 X c» or (a + b)c is the product of the compound quantity a + b multiphed by the simple quantity c, ■ - d ^ I) a + 6 -7-a — 6, or (a + b)-^{a — b)^ or , is the quotient of a + ^ divided by a — b. \/ ab -^ ^d or {ab + cdy is the square root of the com- pound qn-intity ab -p cd, a -f- 6 — c^ or (a + A — c)^ is the cube, or third power, of the quantity a-^- b — c. 5a denotes that the quantity a is to be taken 5 times, and t{Jb -f c) is 7 limes b -f- c. It is also to be remarked that the sig-n + is generally expressed by the word filuH^ or more^ and the sign — by minus J or le/iS. And, in the computation of problems, it must be ob- served, that the first letters ol ihe- alphabet are usually put for known quaniilies, and liie hist lor those which ar*j unknown. 16 ADDITION. EXAMPLES • -*-3a + Sax^ + 60734. 8y +7a + 7flx2 — 3x3+ 7j^ +8a — 3ax« — 13x3+ 8z/ — a — 4ax^ + 2x3— 3z/ —2a + Aax^ + x3— y +9a + \ 2ax2 — 7x3+ 1 9y — 2fl2 + 8A2y3 — 3ad+ r — 3tt2 + 6*2^^3 + 3ad— 10 — . 8^ 10^2^3 + 3aA — 6 + 10a2 20^2^3 — ah^ 2 + I3a2 — b^y^ — ^2ad+ 1 i — 5ary — 8x2y2 — 12x2— 8x — 3xy + 3jr2i/2 + 10x2_3x +8xy —2x2^/2 — \4x^+7x +7a:y +4ar2j/2 + 8x2+ :x; -^^axi — 6\/ax ~2y+2axi -f- flxi + 2y/ax + y+ axi — -Sflxi — ^ t'^ax — 7y— -Saxi +7axi + 10-v/ax +5y+3ax* ADDITION. 19 CASE III. To add quantities tohich are unlike^ and have unlike tigns. y RULE. Collect all the like quantities together by the last rule^. and set down those that are unlike, one aftec another, with their proper signs. 5xy 4ax — xy — 4ax EXAMPLES. 2xy — 1 0x2 2ax — 1 50 +2x^ — 3x2 ^a:y 3x2+2flrx+6x2 — 8x2 — j^y ^xy — 3x*+50 — xy +9x2 ^x+\QO — 5x2 4xy xt/ — 12x2 4ax+4x2+4xy 6x»y2 — 4x2y — ^2<2xy • — 3x2y 12xfl — x2 4«:f+xy 3y2— tfX 2x2—24 6+20^ax — 3y x+ 4^xy+3y y — 2v^ax^— 3y 20+ Zs/ax^^3y .* 3x2y — 3xy« — 3y2x — 8x2y 2v^x — 8y 3v^xi/+IOx 2x+>x+y —8 + y/xy • a^— 8+x*— 2 a — lQ+a2 — jc X*— a* + 8~4 10— a— x2— y [ 20 ] SUBTRACTION. RULE. Change the signs of all the quantities to be subtractec or conceive them to be changed, and then collect th different terms together as in addition. 5a2 — 25 2a2 — 5b EXAMPLES. 6jc2-^ 8v+3 Sixy — 2+ 8jc — y 2jr2+ 9^/— 2 3xy — 8 — 8^ — 3y 3a2+3d 4jc*— .l7i/+5 2arz/+6-f 16x+2z/ Zxy — 8 ^y^—y — 1 ■ — 10 — 8x — 3xz/ — xy+B y^+y+1 xy — 7x+Z — Aioy 4xy — 16 t/2 — 2z/— 2 ■ — 13 — jC — 4xy-^4ay Sx^y — 8 4^xy^--Xy/xy .3x2j/+l 2^xy+2 + xy ^ V 5'ar2-f ^a-— 8— 4^* 6x2—10+43 — x^ oxy — 20 4x^ — 3.(fl+6) xy^ + \Oax/(xy+ 10) 4xy — 30 3f3 — 8.(a-f^) ^2^2+ 2a^\xy-^lO) E 21 ] MULTIPLICATION. CASE I. When both the factor a are timfile quantities* RULE. Multiply the co-efficients of the two terms together, 4nd to the product affix ail the letters in those terms, aud the result will be the whole product required. Xote*, Like signs produce +, and unlike signs — . 12a 3d EXAMPLES. —2a Sa +4b — 6x — 9x —56 , 36ad — Sab - — 30ax +,4,5bx • 7ab -^Sac 6a^x Sx ' _jc2y xy^ — ^y — SSa^bc 30o2x» — x^y^ +7x^y^ — 5ax 3x +5xy —3 — 7xyz — 6a:c" \ 8ax2 Sax Sx^y^ x^y^ — .3j:i/2 \7x^y^> 2ax^ * That like signs make +, and unlike signs — , in the product, may be shown thus : Xst. When + a is to be multplied by + 5 : this implies that -f" " 22 MULTIPLICATION. CASE IL When one of the factors is a corn/iound c^uajitity ^ RULE. Find the products of the multiplier, and every terrr ^ the multiplicand, separately, and place them one tv ^ another, with the proper signs, and the, result will be whole product required. 4(2 — 2 b Sa EXAMPLRS. 6a:y — 8 2x a2 — 2x+6 xy 12a2 — 6ab \2x^y^^\6x u^xy — 2x^y+&xy 13jc— ad 12a S5j>— 7a •X 3y_8+2xv xy 2x^+ay 2xy 12jc2 4y2 — 2.r2 2t/2 — Sx^-^r.v Zxi/^ is to be taken as manv times as there are units in b-y and, since sum of any number of affirmative terms is affirmative, it is plain {+a)X{+b)=-\-ab. 2. If two quantities are to be multiplied together, the i wi]\ be exactly the sa-ne, in whatever order they are pl- a times b is the same as b times a; aiid, the for \ • MULTIPLICATION. - 23 CASE III. ken both the factors are comfiound quaiitities, RULE. iply every term of the multiplier into every term nultiplicand, respectively, and set down the pro- le after another with their proper signs, and their 11 be the whole product required. 5x+4>y 3x — Sy x2-f.jiryJi-»z/2 X — t/ + J,.2 \5x^+\2xy x^-^x^y — xy^ — x^y — xy^-j-y^ ?/+?/% \5x^ + 2xy — 8 1/2 x^ * — 2xy2^-y^' x^+y x^-^y x^+xy+y^ X — y 7/2 x*+yx^ ^^yx^+y^ r x^+x^y-^xy2 .•^x^y'-^xy^ — 1/3 — 1/2 a7*4.2i/x2-f-!/2 X"* * * — y^ a is to be multiplied by -{-b, or +6 by — a, this is the taking — <z as many times as there are units in -j-^; and sum of any number of negative terms is negative, it is hat (— a)x(-f-^) or (-j- 6)x(— fl)=— a^- hen — a is to be muhiplied by — b-. here a — a:=o; ! (fl-r-fl)X — ^ Js also ^o, because o multiplied by any produces o; and since the first term of the product, or* ), is, by case 2, = — ab^ the last term, or (— a)x(— ^), rss-\-ab, in order to make the sum ( — ab^ab)wmo't con- 24 • MULTIPLICATION. EXAMPLES FOR PRACTICE. 1. Multiply 12ar by 3a.' Aub, 36a2jt 2. Multiply 4x2 — 2y by 2y. ^w«. Sx^y— 4i/' 3. Multiply 2A^+4y by 2j:-.-4j/. -in«. 4a;^ — I6y' 4. Multiply x2 — ^J7v4-y2 by jc+y. Ana. x^+y^ 5. Multiply x^+x^y-^xy^+y^ by jc — y. Ana. x* — y^ 6. Multiply x^+xy+y^ by jr2 — ^y+y^, r. Multiply 3a:2 — ^xy+5 by jr^^gjri/ — 3. 8. Multiply 2a2-— 3qx+4x2 by sa^ — eax — 2x^, 9. Multiply 3x3+2jr2i/2+3i/3'by 2x3 — 3x^y^+5y^, DIVISION. CASE L IVhen the divisor is a simfile gtcantiiy, RULE. 1. Place the dividend above a small right line, and the divisor under it, in the manner of a vulgar fraction..' 2. Expunge those letters which are common to both, the dividend and divisor, and divide the co-efficients of all the terms by any number that will divide them without ^ remainder, and the result will be the quotient r^qumd. A'ote*. Like signs make +, and unlike signs*— ^ the same as in multiplication. . ■ ^ -I KJ * That like signs give -{-, and unlike signs — , in tixt qnoti will appear thus : The divisor, multiplied by the quotient, most produce the dividend; therefore. DIVISION. S5 EXAMPLES. I. It is required to find the quotient of a-f-a ; 8dc-^2^; a&d adc-^dcd. . a , 86c ^ - abc a ^««. — .==1; _=s4c; and -7^-. = —. a 2o bed d 3. It is required to find the quotient of l^xy-^^x^^ and (ad+da)-i-2A. 3. Divide }Sx^ by 9jr. ^w«. 24:. 4. Divide lOx^j/^ by — 5x^y. Jna. — 2y. 5« Divide — 9tf:r2[/» by Qx^y. ^n«. — «y. 6. Divide — Bjc^ by — 2x. Jm, +4jc.. 7.' Divide 10a6+15flc by 5a. Ans. 26+3c. 8. Divide 30ax — 54x by 6x. ^n«. Sa^fff. 9. Divide lOx^z/ — l5y* — Sy by Sy. jins^ 2^7*— 3y— I. % . 10. Divide \^a+Zax—\7x^ by 2 la. 11. Divide 3a«— 15+6a+3A by So. CASE 11. When the divisor and dividend are both com/iound quantities* f . Ji* RULE. ' h Range the terms of both the quantities accord- ^ to the dimensions of some letter m them, so that 1 Vfifn both the.terms are -|«, the quotient is -{-, because (-{-) X(^ pfodac ci -|- in the dividend. they are both — , the qqotient is als» +» because (^||}((^} produces — in the dividend. one of them is + and the' other «-, the quotient is --^ {mJ)x(^) produces — in the dividend; and (— }x(— > •1. in the dividend. C i k ^6 DIVISION. the first term may contain the highest power ol letter, the second term, the next highest power ; a on. 2. Divide the first term of the dividend h] first term of the divisor, and place the result in the tient. 3. Multiply the whole divisor by the term thus f and subtract the result from tlie dividend. 4. To this remainder bring down as many ten the dividend as are requisite for the next operatior divide as before ; and so on, as in common arithmet J^ote, If the divisor be not exactly contained i dividend, the quantity which remains after the opei is finished must be placed over the divisor, like a v fraction, and set down at the end of the quotient, coinmon arithmetic. EXAMPLES. J? -f- y)a:2 -[- 2x1/ -}- t/2(x-|- y xy+y^ xy+y^ a+x)a^+Sa^x+Sax^+x\(fi+4ax+gc'^ a'+ a^x 4a*x+4ax^ ax^+x^ ax^-i-x^ 4 DIVISION. 23 X — S)j:3—9:c8+2r;r— 27(0:2— 6ar+9 x^ — 3x2 Sx^+27x -6x2+18jc 9x — 27 9jr_27 ax^'-^x^ ^^y)34— 1/4(^3 + d2y 4- dy2 + y 3 64_63y b^y^ — y4 ^21/8 — by^ by^—y^ by3^y* 2S FRACTIONS. EXAMPLES FOR PRACTICE. 1. Divide a^+2ax+x^hya+a:, Ana.a+x, 2. Divide a^ — 3a^y+3ay^ — y^ by a — y. jin9* a^^^2ay'\-y^. 3. Divide 1 by 1 — x. Ana, X+x+x'^+x^^^c. 4. Divide ^x^ — 96 by 3x — 6. Ana. 2x3+4j72-|-8a:+16, 5. Divide a* — Sa^x+XOa^x^-^XOa'^x^+Sax^ — x« by a« — 2ax+xK Ana. a^ — 3a«j:+3ajr«— x^. 6. Divide 48x3 — 760^2 — 64a2x+ XOba^ by 2x — 3«. 7. Divide y<* — Sj/^x^+Sy^x^ — x« by y3 — Zy^x+Zyxf^ —x\ ALGEBRAIC FRACTIONS. PROBLEM I. To reduce a mixed quantity to an imfirofier fraction, RULE. Multiply the integer by^ the denominator of the ihlic- tioHyland to the product add the nuinerator; and the denominator being placed under this sum, will give the improper fraction required. BXAIIPLES. 5 b 1. Let 3^9 and a-«<- be reduced to improper frac* tions. «5 3x7+5 21+5 26 ^ Firaty 3—. = --U. c= — ^ =x — , Ana, 7 7 7 7 b aXC'^b ac^^b And a—— ss ' ■ ■ ss ■ > Ana, C € C .w FRACTIONS. ^ 3. Let a? H and op-— be reduced to impFo ax ' |cr fractions. ^ . x^ xxa+ar3 ax-\'X^ ^ WiTBtyX H as =s , the Answer, a a a ^no a:— = = ^ ^n*. JT a: or • « ^ 6 . ^ . .62 3. Reduce 8 — to an improper fraction. Ana. — 7 7 fix Q ^JC 4. Reduce 1— — to an improper fraction. Ana, a •^ *" a 5. Let J?— — be reduced to an improper frac tion. 2x— -8 6. Let 10 H ;: be reduced to an improper frac t lion. 7. Let a + r be reduced to an impropei o fraction. X — 3 • •. Let 1 + 2ar— . be reduced to an impropei 5x fl'action. PROBLEM n. To reduce an imfirofier fraction to a whole or mixec itity. RULE. Divide the numerator by the denominator, for the integral part, and place the remainder, if any, ovei the denominator^ and it will be the mixed quantity required. C2 Sff PRACTIOS®. EXlMPt.ES. 17 cw 4- a^ 1. Let -»- and ♦ ' ■■ » be reduced to whole or miKw 5 X quantities. If firsts — =s= l7-T-5aB3f, ^/%^ cmawer required, 5 ax4-a^ a* X ' X ab'-^a'^ ay 4- 2y^ 2. Let — - — and — be reduced to whole o mixed quantities. d o * 35 Sad — ^ 3. Let -^ and ' •■ ■ ' " ■ be reduced to whole or mrxe 8 a 2 ^2 quantities. Jna. 4— and Sb 2. 8 a 4. Let -— - and -— — be reduced to whole or mixe zx ' a— >:r quantities. jf2 y« jj-S y^ . 5. Let J—— and — — be reduced to whole c mixed quantities. 1 Ox^ •— ^ X -I- 3 6. Let - — ■ ' ' ' • be reduced to a whole or mixe 5x quantity. I2x3+3x* T. Let i j g -' ' ' ^1^ - ' be reduced to a whole o mixed quaotitjr^ . * FRACTIONS. SI PROBLEM IIL To reduce Jractiona of different denominatorsi Po others qf the same value^ vfhich Tthall have a common denominator, RULE. Multiply every numerator, separately, into all the denominators but its own, for the new numerators, and ail the denominators together for the common denomi- oator*. • EXAMPLES. ^ a h I. Reduce—^ and — to fractions of equal values be that shall have a common denominator. or xei -Of 6xrasdr ac b^ -— and — s=/raction$ required. be be . 2. Reduce -r-, — , and --- ^^ equivalent fractions^ be d having a common denominator^ axcxd:xsacd bxdXd=:b^d cxpxc^c^b bXcXd^bcd acd b^d ,cH ^ , , , r-v. r-ii and r— , fractions required, bed bed bed j*Whcft the denominators have a common divisor, it will be •^» instead of multiplying by the whole denominators, to multir W only by thefe parts which arise from divi(Uii|^tSLi^^ common / 3S FRACTIONS. 2x b 3. Reduce — and — to equivalent fractions, havin ^ , ^ 2cx a a common denominator. Ana, — and— ac a a a-\-b 4. Reduce — and to fractions, having a con ^ ac ab-^i mon denominator. Ana, -- and — 7- . be be 3jc 2d 5. Reduce — , — 5 and ^to fractions, having a con . 2* ^^ ^ 9cx 4ab ^ 6aa mon denommator. Ana. — — , - — , and --— 6ac 6ac 6a«. 3 ^jc 2^ 6. Reduce -, — , and cH , to fractions, havin 4 3 a . a common denominator. 9a Sax , 1202+ 04 j Ana. — , — —-, and -— 12a 12a 12a 7. Reduce ---> — ' ^^^ — ; — > to fractions havin 2 3 a-^-x a common denom nator. hi* ri 8. Reduce — , — , and — , ^o equivalent fraction; 2a3 2a a ^ having a common denominator. PROBLEM IV. To Jind the greateat common measure of a fraction, RULE*. 1. Range the quantities according to the dimension of some letters, as is shown in division. * The simple divisors, in this rule, may be easily- found, inspection. FRACTIONS. 33 3. Divide the greater term b^ the less, and the last divisor by the last remainder, and so on till nothing re- nuuDs ; and the divisor last used will be the common measure required. Mote, All the letters or figures which are common to each term of the divisors, must be thrown out of them before they are used in the operation. EXAMPLES. 1. To find the greatest common measure of ex+x* or c+x )ca^+a^x(a^ There/ore the greatest common measure is c+x* 3. To find the greatest common me^\ire of x*+%bx+b^)x^ — b^x(x x^+2bx^+6^x —2dx»~2b^x)x^+ 2bx+b^ * or X + b )x^+2bx+b\x+b bx+b^ bx+b^ A 34 FRACTIONS. Therefore x-\-b ia the greateat common aiviaor. J 3. To find the greatest common divisor ©f '. jr« — 1 xy+y 4* To find the greatest common divisor of 5. To find the greatest common measure of a^b+2a^b^+2ab^+b^' PROBLEM V. To reduce a /ration to its lowest terms^ RULE. 1. Find the greatest common measure^ as in the last;^ problem. 2. Divide both the terms of the fraction by tLe CQm-| mon measure thus found, and it will reduce it to its j iDwest terms as was required. EXAMPLES. CX'A'X^ 1. Reduce ^ . ^ to its lowest terms. ca^+a^x ex + x^ya^ + a^x fir c-f-a: )ca^+u^x{a^ ca^^a^x Here, cx-^-x^ ia divided by jr, which ia common to eac ferm. FRACTIONS. Therefore i + «'' «* ^^^ greatest common measur and c+jtV r — a=-r ^fraction reguiret 3. Having -— — - given, it is required to r it to its lowest ter ns :t'2-|- bx bx-\-b^ bx + b2 Therefore x+A is i he greatest common measui X^ ■— i A2 J JC^ __ ff -jQ 3. Reduce^ —--to its lowest terms, jins* — x^'^-b^xf jr2«_7^2 4. Reduce -- — '— t< its lowest terms* Jns, — 5. Reduce -r ; -— - to its lowest terms c3 — a*x-^ ux- f-^-3 0. Reduce -- — —r-- rrT— n — : to us lowest i a^x + '2a^x^-^^:ux^+x^ 36 FRACTIONS. PROBLEM VI. To add fractional quanttUea together. RULE. 1. Reduce the fractions to a common denominator, a in problem the third. 3. Add all the numerators together, and under thei sum write their common denominator, and it will give th sum of the fractions required. EXAMPIES* \, Having — and — given, to find their sum. jrX3=3x xX2=:2jr 3X3=6 3jc 2x 5x 1 = — =s Bum required^ 6 6 6 a c € 2, Having — , — , and -— given, to find their sum. ^ b d / ° axdyf:ii»adj €Xdxf=^cdJ e xbxdsssebd ■l-MBMitaMi«M bxdX/^bdf hdf^ bdf^ b(if bdf ^ ^ FRACTIONS. 37 •3. Let a — — r-" and d+^ — be added together. c « X «t > ^numerators. And bxcsssdc = common denominator, ^ 3x2 ■ aajc 3rj:3 , ^ , Sa^jtr Therefore a }-^H =« 1 ^<^^ — T— = ^ be be be , , . 2abx — 3rx3 . . ^ tf+oH — — e=2 sum required* be , , .,,3ar ,a7 , . I5x+2d.t' 4. Add —7 and — together. Jlna. -^^-r — . 2b 5 lOd X X X .'V -5. Add — • — 1 and — together. jina. x+—., 2 3 4 1 -i 6. Add -- — and — • together. Ans — . 3 7 21 7. Add j:H to 3x-\ ^na. 4x+^ . 3 4 12 tijr2 X \ a ^« It is required to add 4x, -— -, and-- — together, t&d ix fix Tx 2.z*-4- 1 ^' It is required to add --, ^7, and — -7 — togetiur. 3 4 D *0. It is required to add 4jr, -^ , and 2+ -7- lege the ly ^^ It is required to add Sx-] and x ~ together. ^^ In the addition of mixed quantities, it is best to bring the frac ?|^ parts only to a common denominator, and to affix their sw^ to w tvun of the hitegers, interposing the proper sign. D ZS FRACTIONS. PROBLEM VII. To BublrcLct one fractional quantity Jrom ai RULE*. 1. Reduce the fractions to a common dei as in addition* 2. Subtract the numerators from each under their difference write the common dei and it will give the difference of the fractions r EXAMPLES. I, To find the diffcrenceof — - and --:. 3 11 xX11«11j7 2xX 3= 6x 3xlls33 Wa: 6a: 5x ,._ — — — 5=»— =s differ enct 33 33 33 -" 3.^ To find the difference of *-----* and 3d 5c (x— a)X5caB5cx — Sac (2fl— 4x) X 3A SBB 6fl* — 1 2bx ZbX5c^\Sbc Scx-'-^ac 6<7A— 12*x 9jrx— -Sao— 6ad4 \bb~ "^ Tibe • i5cb i^erence required. * ne tame rule maj be observed for mixed ^uan FRACTIONS, 39 ^\2x .3x . 39:c 3. Required the difference of--p and y. ^ns. — . 4. Required the difference of 5y and — . ^ns. — . 5. Required the difference of— and — . ^/i«. -^. d. Required the difference between — ^ and — . tdnSt bd Zx-^a - 2x+7 T, Required the difference of — — - and Ann* 5d 8 24jc+8fl — \Obx — 35^ 406 X • X^'^O ?. Required the difference of 3a: + -— " and a;— . . , cx+bx+a5 Ana.^x-i r- be PROBLEM VIII. To muUifify fractional quantities together » RULE*. Multiply the numerators together for a new numera- ^) and the denominators for a new denominator, and it ^ give the product required. * t When the numerator of one fraction, and the denominator of ^ other, can be idivided by some quantity which is common to each> ^ <IQOdents may be used instead of them. 40 FRACTIONS. EXAMPLES. 1. Let it be required to find the product ol 2. Required the product of—, — , and . ■ * 2 5 21 a: X 4a: X 1 Oa: > 40 jt^ 4x^ S. Required the product of— and . ^ a a + c Sjc Sa 4. Required the product of — and ---. ^ 5. RequH'ed the product of — and . j1 6. Find the continued product of — , — j and a c A 7. It is required to find the product ef />-{ a a Ans, 2. When a fraction is to be multiplied by an integer, thi is found by multiplying the numerator by it ; and if the ii the f»ame with the denominator, the numerator may be take j)roduct. 3. When a fraction is to Jbe multiplied by any quantity, same thing whether the numerator $e multiplied by it, or I minaior divided by it. ' FRACTIONS. 41 d. Required the product of — - — and ■ « 9. Required the product of xH , and — tj« PROBLEM IX. To divide one/ractional quantity by another, auLB*. Multiply the denominator of the divisor by the nume- rator of the dividend, for a new - numerator ; and the Qumerator of the divisor by the denominator of the di^- videndy for a new denominator. Or, which is the same thin^, invert the divisor, and: proceed exactly as in multiplication* EXAMPLES* ♦« Required the quotient of -"-divided by — . X 9 9x 3 -r-X'T-asr— =T= \\ ^a^ quotient required,^ 3. Required the quotient of — divided by —^ Sa d 2ad ad . ^ — X — « —7- = :r7— = quotient required. * 1. If the fractions to be divided have a common denominator; ^^ the numerator of the dividend fior a new numerator, and tte^ "^merator oi the dxviior fbr the denominator. D 3. j. k . I I 42 FRACTIONS. 3. Find the quotient of 7- divided by *^ 2x — -Zb ' Sx+a x+a 5x+a Sx^+^ax+a"^ . ^ ^ .. . /. = — o..« OA - . = 5'"orz>«f required. V.r — 26 «t'-fZ> 2a-2 — 2^2 2r2 X J - 4. Find the quotient of -rr- — divided by ; 2x2 ^x+a ^x^x^x+a) 2x — 7~ — - X = > — : — -= =:nuone: 5. Let — be divided by — . -^^«. -r: fi. Let --- be divided bv 5x. jim, - 7 . • 3 7. Let -r-— be divided by — • ^««. / — i 6 ' 3 4x J? j^ 2 i 8. Let be divided by — . jins. X' I y. Let — be divided bv — . jins. ^ y. Let — be divided by —r, jins. ^ 3 ob %a jc*~""6 3cjr JT"*^ >0. Let be divided by —7 . Ans, — -— 8crf ^ 4rf 6c*x I x^— ^4 x2+dx U. Let — »; • 7- be divided by ^• I x^ — 2bx-^b^ ' X — 2. Wheil a fraction is to be divided by any (quantity, it is the thing whether the numerator be divided by it, or the denominito multiplied by it. 3. When the two numerators, or the two denominators, canb divided by some common quantity, that quantity may be thi«|| out of each, and the quotients used instead of the fractions £ proposedL * [ 43 J INVOLUTION. Involution is the raising of powei*s fram any proposed i^t; or the method of finding the square, cube, biqua- ^fate, 8cc. of any given quantity. RULE*. Multiply the quantity into itself as many times as .^here are units in the index less one, and the last pro- '^Mct will be the power required. Ovy Multiply the index of the quantity by the index of the l^ver, and the result will be the same as before. ^ote. When the sfgn of the root is +, all the powers^ |ofitwili be +; and when the sign is — , all the even. )wcr!i will be +, and all the odd powers — . 0)1 tJ EXAMPLES. 5;-;; I a*=a9guare a^ssicube 'a*ssi4ih power a'=zSth.povifii9 a^ssaguare a^j root > n .^. ^ ' ra^s=s4in power a^^ssSth power I, ^ + 9a^=aguare -^a,root^^ Sia4^^tk power J — 2^3a'^Bs»Sih power. sr u * The nth power of any pradnct is equal to the nth power of JSch of the factors, multiplieci together. (^And the nth power of a fraction is equal to the nth power of the ^H^eratori divided by the nth power of the denominator. 44 1 - fi.T^Ot ^ as biquadrate 40*** -s- agtiarc . tfOOt .3 ^+a«*«5««'' «r sscuB f INVOLUTION. 45 EXAMPLES FOR PRACTICE. 1 . Required the cube or third power of 2a^> Ana, 8a«. 2. Required the 4th power of 2a2 jr. Ane, l^^xK 3. Required the third power of — %x^y^, Ana. — 5l2x6i/0* ~, - 2a^x \6cfix^ 4. To find the biquadrate of -rr» Ana. ^ . , o > ^ 3^2 8168 5. Required the 5th power of c — x. . . Ana, a* — Sa^x+lOa^x^ — lOa^x'-fSax^ — x^. Sia ISAAC NEWTON^s Rule for raiaing a binomial or reaidual quantity to any power whatever*. I % '. To find the terma without the co^effiqienta. The in- *^f ^ of the first or leading quantity begins with that of the ^*^en power, and decreases continually by 1, in every ^^^^ to the last ; and in the following qikntity the in* ^'^es of the temis are 0, 1,2, 3, 4, &c. ^. To find the uncia, or co^efficienta. The first is al- .^*ys 1, and the second is the index of the power : and, '^ general, if the co-efficient of any term be multipUed T }ht index of the leading quantity, and the product be l^^ided by the number of terms to that place, it will give» ^ co-efficient of the term next following. This rule, expressed in general terms, is a|^Uows : a n ^. „«.! n— 1 n— 2, . WPh— 1 #»— 2 n— 3 5B=a -f-n .a 6 + " • "^ ^ 6* + « . -5— . — r- a r^---n n It— 1, . «— 1 n— 2,^ w— 1 n— 3 n^3 o ssa — n.a b-f-n. -^r- fl b^ — «•"■«- • "rT ** 46 INVOLUTION. Kote, The whole number of terms will be one than the index of the given power; and when both of the root are +9 all the terms of the power will I but if the second term he — « all the odd terms will I and the even terms — - EXAMPLES. 1. Let a-\-x be involved to the fifth power> The terms without the co^efficienta will be fl*, a^Xy Q^x^^ a^x^^ ax^f a:*, and the co^tfficitnts will be 5X4 10X3 10X2 5X1 ^nd therefore the fifth power i% «'+5a<x+ iOa3x«+ 10a*j?3+5ax^+x^ % Let x^— a be involved to the 6th power. The tetm^without the co^efficients will be a?*j x*o, jHo>, jc^c*, x*a*, xa^j fl*| and the co^efficientB wiil be 6xS 15X4 20X3 15X2 6X1 h 6, -J-, -j~, -J-, -J-, -g-, or I, 6, 15,20, 15, 6, 1. t^nd therefore the 6M fiower ^o:— ^ « j?a«^jp5a^-i5j;4fla — 20x3a3+l5x2a4— 6ara; ^ Required the 4th power of x— ^. .^n*. x^— 4x3fl+6j?*a**-4j:a' 4. Required the 7th power of x+c. ./^««. j77+7jr6fl+2lj?«a3+35x4a3+35x3a* + 21x 7jra*4-a7, ^ ' ' • ^ . ■ ,LirV ■» The sam of the coefficients, in every power, is equal to th< bar 2, imised to that power Thus, l+laB2 for the first i 14-2+l«4«2» for jAe.squaiej l++33+la:8«s2J f euM^ or thicd power^||B/ - • J J £ 47 ] EVOEUTION. Evolution is the reverse of involution, or the method ^ finding the square root, cube root, Sec* of any given ^IKiantityy whether cimple or compound. CASE I. To find the roots of simple quantities. RULE*. Extract the root of the co-efficient for the numerical put, and divide the index of the letter^ or letters, by the index of the power, and it will give the root required. EXAMPLES. 1* Required the square root of 9^S And the cube root of 8x3. ^no* v^9^'asSjr|aB3x; and •^8:p^83x|sa3x. . ^ 3x*y* 3. Required the square root of ' » and the cube root 27a^ 3r«f^ xy ^^ , A6x^^ 2xy ^. - * Any even root of m affinnative quantity may^ be ekher + «.— {- thus the square root of ^ «t it either ^ a, or i— a : for (+«)X(+fl)— +a». «nd (-.fl)x(— «)— +A And an odd- root of any quantity will have the same eign as fhc q^ianidty ittk^: thw the onbe root of ..\ufi it ^randtho eube y^ 48 EVOLUTION. 3. Required. the square root of 3a3x<^. Ann^ax^y/ 4. Required the cube root of — 125a3x^» An9. — 5a j 5. Required the square root of — j- . An9» ^ — = 6. Required the 4th root of 3560^x8. Ans. 4a:i 7. It is required to find the 5th root of — SSx^yi®. Ana. —2x1 CASE II. Tojind the square root of a compound qttantity, RULE. 1. Range the quantities according to the diraenaio of some letter, and set the root of the first term in tl quotient. 2. SubtracUthe square of the root, thus found, fro the first term, and bring down the two next terms to tl remainder for a dividend. 3. Divide the dividend by double the root, and set H result both in the quotient and divisor. 4. Multiply the divisor, thus increased, by the ter last put in the quotient, and subtract the product fro the dividend ; and so on, as in common arithmetic. Trootof — a3 is -^; for (+a)X(+fl)X(+fl)=«+fl^and(-< Any even root of a negative quantity is impossible : for neftH {4-a)X(+«) nor (— «)x(— «) can produce — . The nth root of a product is equal to the nth root of each ff fjj factors multiplied together. « And the nth root of a fraction is equal to the nth root of th»f| joentor, (fa'vided Iby the nth lOot of the denominator. ' . ^ EVOLUTION. 40 EXAMPLES. 1. JBxtract the square root of x^— 4jc3+6jJ'aL--4:c+I. 074 — 4^:3+6^:2 — 4:c+l(x2— 2a:+l=: root. — 4jr3-f^4j7* 2x^ — 4x+ 1 2. Extract the square root of 4a4+ \2ii^x.+ ISa^ir* 4fl4-{- i2a3jK.-f I3a2j.2-|-6ax3+x4(2a«+3(rx+x2 4fl4 4c^+ 3aar) 1 2c«a: + 1 3a>:r3 42a3x+ 9a2x8 4o2+6ax+x2) 4ta^x^+6ax^+x^ 3. Rcquu'ed the square root of fl'* + 4a3x + 6a?a:* +Aax^+x^. Jina. a^+^ax,+x^. 4. Required the square root of x^o-^x^+^x^"^ X ^ 2 ^16 ^ 5. It is required to find the square root of a«+ x^. 2a 8a3 vl6a* E 50 EVOLUTION. CASE III. Tojind the roots o/fiowera in general, RULE*. 1. Find the root of the first term, and place it quotient. 2. Subtract its power from that term, and bring the second term for a dividend. 3. Involve the root, last found, to the next ] /power, and multiply it by the index of the given for a divisor. 4. Divide the dividend by the divisor, and the tient will be the next term of the root. 5. Involve the whole root, and subtract and as before ; and so on till the whole is finished. EXAMPLES. 1. Required the square root pfc^ — 20^^17 + 2a«)— 2o3jr a*— .SflSjc^ c2x2 2a«)2aax« a^ — 2a^x+Za^x^ — 2ax^ + x^ As this ni£thod, m high powers, is generally thought tc !!^ll "*^ "^' ^ improper to observe, that the roots of cor I EVOLUTION. 5 1 2. Extract the cube root of x»+6x«— 40x3+96x — 64. a:« + 6x« — iOx^+96x — 64(a;2+2x— .4 3x4) 6x* x«+ 6x5 +12x4+8x3 3a74)««i2x4 x«+ 6x«— 40073^+ 96 jp— 64 3. Required the square root of ««+2a3+2flc+42 +2bc+cK jina^a+b+c. 4. Required the cube root of x^ — 6x«+15x<— -20 ar5+}5x»— 6x+l. .^n«. xa— 2x+l. 5. Required the biquadrate root of 16a4 — 96a^x + 2 1 6a2x2— 2 1 6ax3 + 8 1 x^. Jn8. 2a— .3x. 6. Required the" fifth root of 32x« — 80x4+80x3— k, 40a:2+iOx — 1. Jns. 2x— 1. 1- Extract the roots of some of the most simple tenns, and con- 'Jttit them together by the sign + or — , as may be judged most ^iritable for the purpose. 3 Involve the compound root, thus found, to the proper power, ^^^» if it be the same with the given quantity, it is the root re- quired. 3. But if it be found to differ only in some of the signs, change ^^m from + to — , or from — to +, till its power agrees with the given one throughout. Thus, in the fifth example, the root 2a — 3x, is the difference (jf the roots of the first and last terms ; and in the 3d example, the root a-^-b-{'C is the sum of the roots of the Ist, 4th, and 6th terms. The same may also be observed of the 6th example, where the root i» found from the first and last terms. [ 52 ] SURDS- Surds are such quantities as have no exact root, be usually expressed by fractional indices, or by mean the radical sign \/ placed before them. Thus, 2', or \/2, denotes the'square root of 2, anc the cube root of the square of ^ ; where the numen shows the power to which the quantity is to be rais and the denominator its root. PROBLEM I. To reduce a rational quantity to the form of a surd RULE. Raise the quantity to a power equivalent to t denoted by the index of the surd, and over this i quantity plaee the radical sign, and it will be of form required. EXAMPLES. 1. It is required to reduce 3 i% the form of square root. \,Firat^ 3x3sbs3*=9; vyhenccs/^ the ansz 2. It is required to reduce 2x« to the form of cube root. I^irttt^ 2arX2j7»x2a:2=:(2x'>)3=8xS; whence ^8x (Sx*) !» the answer » 3. Reduce 5 to the form of the cube root. Jna, (125 4. Reduce hxy to the form of the square root. SURDS. 53 1 5. Reduce 2 to the form of the 5th root« ^na, (32)^. 1 6. Let d^ be reduced to the form of the 6th root. 7. Reduce a + b io the form of the square root, and a — 6 to the form of the cube root. PROBLEM ir. To reduce quantities of different indices to other equiva- lent ones that s/iall have a common index. RULE. 1. Divide the indices of the quantities by the given index, and the quotients will be the new indices for those quantities- 2. Over the said quantities, with their new indices, place the given index, and they vill make the equivalent' quantities required. 3. A comtnon index may also be found by reducing the indices of the quantities to a common denominator, and involving each of them to the power denoted by its Dumerator. EXAMPLES. 1. Reduce 15* and 9^^ to equivalent quantities having the common index i. 1 1 1 2 2 1 . , . — -» s — X — = — « — «Ui index* 4 2 4 1 4 a 1 1 1 2 2 1 «... L.^Bs X — sss— OB — n2(/ index*. 6 2 6 1 6 3 Thertfore{\5^)^ and (9^)*— quantiHcM re^uire<t:^ R 2 54 SX^RDS. 2. Reduce o^ and x-^ to the same common index ^ "2*3 1^1 I 5 = — X — ==— =2a index, 4 3 4 1 4 1 1 Therefore {aP'^ and (^-fp = quantities required, * 3, Reduce 3 and 2"^ to the common index -J. I Ans, (27)'^ and ( 4. Reduce a* and 6^ to the common index |. 1 Ans, {'i^j^ and (.' i , 1 5* Reduce a** and A"* to the same radical sign. Ans. "^x^a"* a?id '""v i ^ 6.. Let (<! + *) and (a— *)J be reduced to a comi index. 7. Let (a + ^)^ and (a*— d)* be reduced to a comi index. PROBLEM IIL To reduce aurda to their moat aim/ile terms. RUI/E*. Find the greatest power contained in the given si and set its root before the remaining quantities, uith proper radical sign between them. * When the given surd contains no exact poweri it is alread MtM mam tkB^mmx SURDS. 55 EXAMPLES. 1. It is required to reduce ^/48 to its most simple terms. v/48=v/(l6x3) =?v/ 16 Xv/3=4X x/Srsfv^^, the answer. '2. It is required to reduce -^108 to its most simple terms. ^VIOS =:^ (27X4) =^27 X^K4 = 3x ^4 =3 ^4, ■'/fcf answer. 3. Reduce \/125 to its most simple Jt^rms. 50 4. Reduce \/ t—— to its most simple terms. 147 5. Reduce -^243 to its most simple terms. ^/i*. 3^9. 16 6. Reduce -^ — to its most simple terms. 81 * Jns. 1^18, 7. Reduce x/9Sa^x to its most simple terms. 8. Reduce ^(x^— a»jp») to its most siipple terms. 9. Reduce (fl3jc+3a3jr«>J to its most simple terilRS. 10. Reduce (32afi — 96a^a;)J to its most simple terms. 3* 56 SURDS. PROBLEM IV. To add surd quantities together. ^ « ^ RULE. 1. Reduce such quantities as have unlike indio ether equivalent ones, having a common index. 2. Bring all fractions to a common denominatofi reduce the quantities to their simplest terms, as ii last problem. 3. Then, if the surd part be the same in then afiuex it to the sum of the rational parts, M'ith sijg;n of multiplication, and it ^ill give the total reqAiired. But if the surd part be not the same in all the qt ties, they can only be added by the signs + and — . EXAMPLES. 1. It is requh*ed to add y/27 and ^48 together. Firsts -v/27=v^(9X3)=3-v/3; owcf ^^48 =v/( a)=r:4v'3; Whence, 3v^3+4v/3=(3+4)^3»7v^3=i aui quired. 2. It is required to add -^500 and ^108 togethe Firat, v^ 500 =^ ( 125 X-4) ssS^4, and^ 108 » (27X4)«3^4; fVhencej 5^4+3^4w(5+3)^4»8^4«, ««« quired. SURIJp. . 57 3. Required the sum of ^72 aiid ^/128. Ana. llv^2. 4, Required the sux^^jf v^ 2 7 and ^147. Ana. lOV'S. ' An8,\iy/6. ^^j^biA- «i^/» 6. Reauirw^ tl^ sum of ^40 and -^135, 7. Required the sum of 3^1 and S^V^V* 8. Required the sum of l^a^b and %^^Xbx^. 9. Required the sum of 9^/243 and 10v^363. 10. It is required to find the sum of a** and a"» . 1 1. Required the sum of ^Ita^x and y/Za^x, PROBLEM V, To auhtract^ or find the difference of^ surd quantiUes. RULB. ' Pi'epare the quantities as in the last rule, and the diffe- rence of the rational parts annexed to the common sard will give the difference of the surds required. Hut, if the quantities have no common surd, they can onlr be subtracted by means of the bign — s 58 * Sd^RDS. EXA.MPLES. 1. It is required to find the difference of v^.448 and Firsts V'448 = %/(64 X 7) Si^r ; and v' 1 12 = v/ Whence 8v>^l^4x/7. = (8« — •*)v'r = 4vr'7, atjperencc ^ required. 1 2. It is required to find the diffwrcnoc of lor^^ nnd 24* ''^^ Firsty 192^= (64X3)*^ =4.3*; flwtf 24* « (8X3)^=s " ' 1 2.3^. 11 11 Whence 4.3^— 2.3^ = (4 — 2)^.3 == 2.3'3^=x: difference required* 3. Required the difference of 2v^50 and v^l8. An9, 7-v/2. 1 1 4. Required the difference of 320^ and 40"^, Ana, 2 ^5 . 3 5 5. Required the difference of \/— and %/--. 6. Required the difference of ^-f and ^-5^. 7. Find the difference of y/^Oa^x and ^/20o2x3. ^/i*. (4a2— 2(w?) \/5:c. 8. Required the difference of S^a^A and -^a^b. 1 1 9. It is required to find the difference of x^ and x"* . SURDS. 59 PRjgBLEM VI. To multijily surd quantities togtAler. ^f^^ RULE. Reduce the surds to the same index, and^fteproduct efthe rational quantities annexed to the pit>duct of the ^ surds will give the whole product required ; which may be reduced to its most simple terms by Problem 3. EXAMPLES. 1. It is required to find the product of 3 V' 8 and, i Here, 3 X2 X v'SX v'S =6^/(8X6) = 6 v^48:=6 v" (16X3)=6x4X\/3=24v/3= /irorfwc/ required, 2. It is required to find the product of ^^f and 13 2 5 3 .10 3 ^15 5 I 3 1 X^IS = — ^ 1 5 = - ^ 1 5 = firoduct required. 3. Required the product of 5v/8 and 3-^/5. jins, 30^/ It). 1 2 4. Required the product of — \/6 and— ig^lS. ./fw«. -^4. 2 1 3 7 i^. Required the product of -—v/—- and -7-^/— . 60 SURDS. / 6, Reqviired the product of -^18 and 5-^4. 7. It is reqjiiced to find the product of <z* and a "3^. - • An8*{a^yor 1 1 8. Required the product of (jt+j/)^ and (^x+y)"^. 9. Re<K|iABd the product of x+^y and x — ^y« 10. Required the product of (a+\/^)^> and (a- JL 1. 11. It is required to find the product of x« and x^. PROBLEM VII. To divide one surd quantity by another, RULE. Reduce the surds to the same index, and the quotie of the rational quantities being annexed to the quotie of the surds, will give the whole quotient requirec which may be reduced to its most simple terms before. EXAMPLES. 1. It is required to divide 8^108 by 2^6, (8-5-2) v'(108-^6)=4^18=4v^(9x2) =4x3^/2= 12^/295 quotient required. 2. It is required to divide 8^512 by 4^2. Ill 1 > 8-i-4=a:2, and 512' H-2'^=:256"^sB=4.4'^ ; Therefore 2x 4x4'y« 8.43= quotient required. ^. Let €%/lO be divided by 3>/5, An9.2y/i SURDS. 61 4. Let 4^1000 be divided by 2^4. Ana, 10^2. 3 1 2 1 1 5. Let— v/—- be divided bx — \/— . Ana, -r-v^3. 4 135 ,35 o K *i 2 3 25 6. Let --^— be divided by — ,3/—. Ana ^3. 7 3 '54 21 2 2 1 31 8 > 7. Let — \/a. or — a^. be divided by — a"^. •^w*. — a"*^. 5 5 ' 4i '%. 15 8. Let the quantity x^ be divided by the quantity x^ . 9. Let x* — ^jcrf — b+di/b be divided by x — ^/b. PROBLEM VIIL To involve aurd quantitiea to any fiotoer. RULE. Multiply the index of the quantity by the index of the power to which it is to be raised, and to the result annex the power of the rational parts, and it will give the power required. EXAMPLES. 2 1 1. It is required to find the square of -—a''^. Firat, r±)2^±x — = — , and Ca'^y=za'^ '^ =a^ 3 3 3 9 ^ 2 1 4*4 Whence (— a^)* =*"^^*)^ =* — ^a* = aguare requirei. o y «f / 62 SURDS. 5 It is required to find the cube of — v'''* FiTBU (1)3= Ix-i xf =^, and (7^)3 =7^ =(73)5; 5 12^ ^ 125 ^ M^A^wcf (— v^7)3 = — • (73)^ = — • (343)^ ^cube re^ quired, . 3. Required the square of 3^3. ^n*. 9-^9. 1 4. Required the cube of 2^ or -v/2. ^n«. 2^2. 1 * I 5. Required the 4th power of — v'6. -— . 6.x 36 6. It is required to find the wth power of a*" . 7. It is required to find the square of 3 + V'S. 8. It is required to find the cube of 2x — 3^y. PROBLEM IX. To extract the roots of surd quantities^ ' RULE*. Divide the index of the given quantity by the index of the root to be extracted, and to the result annex the root of the rational part, and it will give the root re- quired. • The square root of a binomial or residual surd a + b, or A — B, may be found thus : take \/( a^ — b^) = d ; then \/{a. Thus the square root of 8+2\/7=l+ \/7, and the square root of 2 — \/8=\/2— 1; but for th? cube, or any higher root, no general rule can be given. SURDS. . 63 EXAMPLES. 1. It is required to find the square root of 9^3. 1 1 WhencCyi^^Z^isi^.^ ^s^square root required, 2. It is required to find the cube root of — v^2. First, ^-5-=-, aTzrf (^2)^=2^ ' =2^; 1 I- 1 1 ' Whence^ ( — ^2) = — 2^ =:cube root required. 3. Required the square root of \0\ Ana. 10^/10. 8 2 4. Required the cube root of ;—a3. Ana.^, 1 1 5. Required the 4th root of 3jc^. ./fn«. 3^. x^. X ^ 6. It is required to find the 72th root of x»». 7. Required the square root of j?* — 4iXx/a+4ia. INFINITE SERIES. jtn irifinite aeries is formed from a vulgar fraction, having a compound denominator, or by extracting the root of a surd quantity ; and is such, as, being con- tinued, would- run on ad infinitum, in the manner of a decimal fraction. But, by obtaining a few of the first terms, the law of the progression will be manifest, so that the series may be continued) without actually performing the whole operation. 64 INFINltE SERIES. PROBLEM I. To reduce fractional quantUlea into injinite series*. Divide the numerator by the denominator, as in common division; and the operation continued, as far as may be thought necessary, will give the series re- quired. EXAMPLES. ax . . . . . a — X uc piupuac ;u Lvi UC l«UIIY< infinite series. , a-— a:)ax...( x-\- a a^ xA ' a3' &c. aor— -X» 9 X^ / X^- ^3 ■ a x^ a x^ x^ a ■u» 07* . a» 07* X* - ♦ a*"" ■^o^ x^ rNFINITE SERIES. 65 2. Let — : — be converted into an infinite series. 1+X)1 (1 X + X^ X^y &c. l-i-x 'X — x^ x^ x^-{-x^ 'X^ . -x^~^x^ x^ I . Let be converted into an infinite series^ a-^x b ^ x^ x^ An8.—X^\ +— r+, &€.>; a a a^ a^ 4. Let be converted into an infinite series. a — X hi X x^ \ Jins. — x( 1+— +-^» &C.1 a \ a a^ / 5. Let be converted into an infinite senes. 1 — X a* 6. Let r -^ be converted into an infinite series* jina. 1 +-r 7»^^- a a* «t* r. Let -, or its equal be converted into an inv- 2 ^ 1+1 finite series*. E2. 6 INFINITE SERIES. PROBLEM 11. To reduce a compound surd into an infinite series. .auLB*. Extract the root as in common arithmetic, and the operation 9 continued as far as may be thought necessary will g;ive the series required. EXAMPLES. I. Extract the square root of a^+x^ in an infinite series. ^, . a:« x^ , x^ Sx^ ,_^ ^ ^ ^2a 8o3^ I6a« 1280^' 2a+— )x8 2a' 2 1 ^^ ^ 4a2 x^ x^ x^ X* X* X"' 0«il*t 4a2 Sa-* • 64a» x* x^ ^ ^ x^ x^ 2a+ — — TT) ^^0 4a3' ' ga-* 64a<» x^ x^ 8a* 16a» 5x8 64fl«' * This rule is chiefly of o^e in extracting the square r apention Iwing too tedious for the hi£;her powers. ^ INFINITE SERIES. 67 2. Let ^^ 1 + 1 be converted into an infinite series. 1111 '2 8 ^ 16 32 3. Let \/fl2— .x* be converted into an infinite series. 07* X* x^ ,_, 2a 8a3 i6a« 4. Let -^1 — wc3 be converted into an infinite series. . ^ x^ ^fi^ Sx^ ,_ 3 9 8l' 5. Let ^/a^+b be converted into an infinite series. PROBLEM III. ^0 reduce a binomial surd into an injinite series ; or t9. extract any root of a binomiaU RULE*. Substitute the particular letters of the binemial with f^eir proper signs, in the following general form, and \ ^JH give the root required; observing that p is ^"G first term, q the second term divided by the first, m ^ the index of the power or root \ and a, b, g, d, &c. the *o^cgoing terms with their proper signs: ' -f qp« = P„ (a) + -jj- Aq(B) +-^ Bq(c) + — 3^ Cq(D) + !:iZl5:?Bq(B),&c. Any surd may be taken from the denominator of a fraction and '^ in thi index. f}^^ in the numerator, and vice versa, by only changing the sign of. 68 INFINITE SEKIES. EXAMPLES. 1. To extract the square root of r^ — x^, in an infinite series. / x^ m 1 Here P=srS qs=:_— -, and — = — ; therefore (r* — -'7=r+ (± X A X-f!) + (-^ X » X -5) + (- _XCx--j + (- JXDX--J, t^c. = r+ (__ A H rB H -c +'r-rD> ^c. which, by restoring^ the 4r^ 6r* or* x^ x^ x^ values o/aj b, c, d, ^c. becomes ^— ^j; r-^- — -rT^ — 5078 l28r^ , Isfc, = series required. 2. To find the value of ; — ; — -, or its equal (a + b)-^^ (a -f- d)^ in an infinite series. b m —2 Here p=b a, qs=— , and — = ; therefore (a + by*' --+(-f'4)+(44)+K4)+(- f 5b \ .. \ 2b 3b 4b 5b ^^ ^\ 4a r a^ a 2« 3a 4fl * which, by restoring the values of a, b, c, d, becomeM: 1 23 3d 4b sb . ^ . ^ a2 o3 o* tt« ^ao • ^ INFINITE SERIES. 69 3. To find the value of — : — , in an infinite series. jins. r — xA 1 , ^c. i. To find the value of-- — '-^i in an infinite series. I x2 3x4 \5x^ _, 5. To find the value of;: — ; — -- in an infinite series. 2x 3x2 4j;3 5j;4 6. To find the value of (a*+*)^ in an infinite series. b b^ b^ 564 ^2a 8a3 lea* 128a7' 7. Find the value of (a^ — x2)T in an infinite series. •*»•• "^ X ' 1 - t:;!- 21;?- i2ir»' ^'- 8. To find the value of (a^ — ^)^in an infinite series. b b^ 5/>3 1064 jins.a — —^ — ——^^^ — .^^j5^ , l:fc . 9. Required the square root of — JL — m an infinite series. " •* X* X4 X® a^ 2a4 "^ 2a0 10. Required the cube root of — f in an in- finite series. (as+x^)^ 1 . 2x* 5x4 40x« ,^ Ana. — ;X : 1 1 ^> ^f- of 3a2 ^9tt4 81a6' 70 ARITHMETICAL PROPORTION. 1 1. Required the value of ^ in an infinite series. " ax-^x ^ X :r2 ^4 ^5 ^ne, j iSfc* a a^ a* a* ARITHMEICAL PROPORTION. Arithmetical firofiortion is the relation which two quan- tities, of the same kind, bear to each other, with respect to their difference. Four quantities are said to be in arithmetical firofiortioriy when the difference between the first and second is equal to the difference between the third and fourth. . Thusy 3, 7, 12, 16, and a, c+A, c, c+6, are arith* metically firofiortionaU Arithmetical firogreaaion is when a series of quantities either increase or decrease by the same common differ- ence* Thus^ 1, 3, 5, 7, 9, 11, b'c. and a, a-f-^i a'\-2by fl+3^, a+4d,^a+5A, iSJ'c, are series in arithmetical fir o» gression^ whose common differences are 2 and b. The most useful part of arithmetical proportion is contained in the following theorems : I. If four quantities be in arithrhetical. proportion, the sum of the two means will be equal to the sum of the two extremes. Thusy if 2^ 5, 7, 10, a?id a, d, r, d^ are in arithmC' tical ftrofiortiony then will 2 + 10=5+7, and a+e/ = b+c. II. In any continued arithmetical progression, the sum of the two extremes, and that of any two terms which are equally distant firom them, are equal to each other. ARITHMETICAL PROPORTION. 7 1 Thu8^ in the aeries 2, 4, 6, 8, 10, 12, e^c. 2 + 12 = 4 + 10=6+8. III. The last term of any arithmetical series is equal to the sum or difference of the first term, and the product of the common difference by the number of terms less * ODe ; according as the series is increasing or decreasing. Thua^ the 20th term of 2, 4, 6, 8, 10, 12, ^Tc. m =2 + 2(20— .l)n=2+2x 19=2 + 38=40. And the nth term ofa^a'-^Xy a — 2a:, a — 3x, a — 4^, ^cia ssa — (n — l)Xx=a — (n — l)x. IV. The sum of any series of quantities in arithme- tical progression is equal to the sum of the two extremes multiplied by half the number of terms. Thuay the sum o/* 1, 2, 3, 4, 5, 6, ^c. continued to the (1+20)X20 21X20 ^, ,^ ^,^ 20th term^ia =i — = — r — = 21x10 = 210. And the lum ofn terma of a^ a + x^ a -{- 2x, a + 3x, to a-\-mxy ia ss (a+a+mj:)-—=(a+^mar). w = (aH — •— x)n. EXAMPLES. , I. The first term of an increasing arithmetical series is 3, the common difference 2, and the number of terms 20 ; required the sum of the series. /^ir«^,3+2x(20—l)=3 + 3x19=3+38=41= aat term, And(3 + 4\)X — = 44X — = 44X 10 = 440= sum ^2 2 Required, 2. The first teiin of a decreasing arithmetical series is 100, the common difference 3, and the nun)* " ms 34 ; required the sum of the series. 72 ARITHMETICAL PROPORTION. Firaty 100—3.(34 — 1) = 100 — 3.(33)= 100 — 9 1 = last term. 34 34 ^«rf(l00+l)x~ = 101X--=101 X 17=17n sum required, * 3. Required the sum of the natural numbers 1, 2 4, 5, 6, Sec. continued to 1000 terms. AnB. 500< 4. * Required the sum of the odd numbers 1, 3, 5 9, Sec. continued to 101 terms. Ans* 102 5. How many strokes da the clocks of Venice, wh go on to 24 o'clock, strike in the compass of a day? Ana, 3 6. The first term of a decreasing arithmetical serie 10, the common difference -|, and the number of tei 21; required the sum of the series. Ana. 1 7. One hundred stones being placed on the ground a straight line, at the distance of a yard from each otl how far will a person travel who shall bring them one one to a basket, which is placed one yard from the f stone ? Ana. 5 milea and 1 300 yai • The sum of any number of terms (n) of the arithmetical se of odd numbers 1, 3, 5, 7» 9» &c. is equal to the square (n^) of t number. That is, if 1, 3, 5, 7, 9, &c. be the n)Knbers, Then will 1, 2^, 32, 42, 5*, &c. be the sums of 1, 2, 3, Stc. tern For 0+li or the sum of 1 term =1*, or 1 1+3, or the sum of 2 terms =2*, or 4 4-f-5, or the sum of 3 terms =3*, or 9 9-f 7, or the sum of 4 terms =4*, or 16, &c. Whence it is plain, that, let n be any number whatcveri the s of/i terms will be rfi. [ 73 ] GEOMETRICAL PROPORTION. Geometrical. flroflortion is that relation of two quantities of the same kind, which arises from considering what part the one is of the other, or how often it is contained 10 it. # When four quantities are compared together, the first ~ and third are called the antecedentsy and the second and i' fourth the conaequents. Ratio is the quotient which arises from dividing the antecedent by the consequent, or the consequent by the antecedent. Four guantitiea are said to be flrofiortional when the first is the same part or multiple of the second as tl^ third is of the fourth. Thus 3, 8, 3, 12> and a, ar, *, br are geometrical firo- /i^rtionala. . Direct firofiortion is when the same relation subsists between the first term and the second, as between the third and the fourth. Thfis 3, 6, 5, 10, and x, axj y, ay are in direct firo^ Mrtion. Reciprocal or inverse firofiortion is when one quantity increases in the same proportion as another diminishes. Thuaj 2, 6, 9, 3, and <i, ary br^ b are in inverse firo- fiortion. A aeriea of quantities are said to be in geometrical ^Progression when the first has the same ratio to the ^cond' as the second to the third, the third to the fourth, &c. Thusy 2, 4, 8, 16, 32, 64, Isfc. and fl, ar, or^, ar^ ai*, <ir', t3*c. are aeries in geometrical fir ogreaaion. 74 GEOMETRICAL PROPORTION. The most useful part of geometrical proportion is con- tained in the following theorems. I. If four quantities be in geometrical proportion, the product of the two means will be equal lo that of the two extremes. Thu8y if^y 4, 6, 12, and a, ar^ by br, be geometrically firo/iordonalj then vnU 2x 12=as4X6, and axbrz=bxar. II. If four quantities be in geometrical proportion j the rectangle of the means, divided by either of the extremes, will give the other extreme, Thusf ifZy 9, 5, 15, and Xy axy y, ay^ are geometrically firofiortionaly then will = 15, and . ss=x. 3 ay III. In any continued geometrical progression, the product of the two extremes, and that of any other two terms, equally distant from them, will be equal to each other. Thus, in the aeries 1, 3, 9, 27, 81, 243, i^Tc. 1 X 243 = 3X81 = 9X27. IV. In any continued geometrical series, the last term is equal to the first multiplied by such a power of the ratio as is denoted by the number of terms less one. Thusj in the series 2, 6, 18, 54, 162, ^c. 2x34=162. V. The sum of any series in geometrical progression is found by multiplying the last term by the ratio, and dividing the difference of this product and the first term by the ratio less one. Thusy the-sum of *^y 4, 8, 16, 32, 64, 128, 256, 512 is 512x2 — 2 -—=1024—2=1022. 2 — 1 And the sum of n terms of a, ar, ar^^ oH, cH, ^c, to ar"~^Xr — a ar^ — a r**— 1 GEOMETRICAL PROPORTION. 75 VI. If four quantities, a, 6, c, e/, or 3, 6, 5, 15) be pro- portional, then will any of the following forms of those quantities be also proportional. 1 . directly^ a : b i : c : d, or 2 : 6 : i 5 i 15» 3. inversely^ d : a : : rf : r, or 6 : 2 ; : 1 5 : 5. 3. alternately^ a : c : : 6 : rf, or 2 : 5 : : 6 : 15. '^■ 4. com/ioundedly, a : a+b : : c : c+c/, or 2 : 8 : : 5 : 20. 5. dividedly, a i b—ra ::c : d-^c^ or 2 : 4 ; : 5 : 10. 6. mixed^ b+a : A— a : : d+ci d-^Cy or 8 : 4 : : 20 : 10. 7. by multi/ilication, ro : rd : : c : </, or 2.3 : 6.3 : : 5 : 15. 8. by division^ a-i-r : b-r-r : : c : rf, or 1 : 3 ; : 5 : 15. 9. The numbers a^ by c^ d are i0ltarmonical /trofiortion, when a : diiauibi c Vid, EXAMPLES. 1. The first term of a geometrical series is 1, the ratio 2, and the number of terms 10; whait^ is the sum of the • > . .j . series f Fir9ty 1 X2«==» 1 X5 12= /a«r term. .^512x2—1 1024—1 ,^^^ . . jind = = 1023 = aunt required. 2 — I 1 2. The first term of a geometric series is ^ the ratio ■|, and the number of terms 5 ; required the sum of the series. Fir>.t, 1 X (-1)' =1 X ^ = 4 - last term. •^"'' (i—rk^i-)^ ('-- i-)-(T-i56)-^T 121 3 121 121 . . =: X — = = ^ aum required* 243 2 81X2 162 3. Required the sum of 1, 3, 9, 27, 81, &c. continued to 12 terms. -^^w. 265720. 76 SIMPLE EQUATIONS. 4. Required the sum of ^ 7) •}> ^) Tt> ^^' continued to 13 tenns. 5. Required the sum of I, 2, 4, 8, 16, 32, &c. conti- nued to 100 terms. SIMPLE EQUATIONS. An equation is, Vfht% two equal ^antities, di fferently expr^pd, are compared togetheTEy means of the sign ass plac^Rfetween them. Thus 12 — 5=s7 is an equation, expressing the equality of the quantities 12— -5 and 7. A aim/lie equation is that which contains only one un- known quantity, without including its power. Thus X— a+^ssc is a simple equation, containing only the unknown quantity x. Reduction if equations is the method of finding the va- lue of the unknown quantity, which is sho\m in the fol- lowing rules. RULE I. Any quantity may be transposed from one side of the equation to the other by changing its sign. Thus J i/'x-f 3=7, then will x=7— 3=s4« Andf if X — 4 -f 6 = 8, then will j? = 8 -f 4—6 a= 6.. Also^ (f x-^-^a+bsssC'-^dj'ihen will x=:c — t/-f c— d, Andf in like manner^ (/'4x— 8=3j:-f-20, then will 4.r .«.3x=:30-f 8, or a:=28. RULE II. If the unknown term be multiplied by any quantity, it may be taken away by dividing all the other terms of the equation by it. SIMPLE EQUATIONS. 77 Tku8j ifax-ssab^-^y then will xas^^-— 1. jindj i/" 207+4=16, then wiil a:+2»Sj and xsbQ.^2 = 6. In like manner^ if ax + 2ba = 3c», then 'mill x^2b b^. — • and x^ssi 2d. a a HULE III. If the unknown term be divided by any quantity, it may be taken away by multiplying all the other terms of the equation by it. Thusy (/•^=55+S, then wz7/ 07=10+6=16. X ^4ndj if — =d+c— </, then wiU xssab+ac — ad. a 2x In like manner j if 2=6+4, then will 2x — 6=18- 36 + 12, a«c/2x= 18+ 12 + 6=36, or x^ — = 18. RULE IV. m The unknown quantity in any equation may be made free from surds, by transposing the rest of the terms by Rule I, and then involving each side to such a power as is denoted by the index of the surd. Thuay ifx/x — 2= 6, then will s/x = 6 + 2 = 8, and x=8a=64. ^«rfi/\/4x + 16=12, then w«7/ 4a: + 16 = 144, or ^x = 144 — 16=128; and if both sides of the equation be divided by 4, x will 6e s= 32. G2 ra SIMPLE EQUATIONS. In like manner^ «/-^2jr+3+4:5=8, then Vfill -^2. »8 — 4=4, and 2j7+3»43s64, and 2.r^64 — 3; 61 or x^T =30i. «|- 2 '^^' RULE v. If that side of the equation which contains the known quantity be a complete power, it may be r^d by extracting the root of the said power from both of the equation. 7%tt», «/x«+^x+9=25, then will x+3ses^25 = jindf r73:c«— 9=21+ 3, then will 3a?«=21+ 3 H 33 S$^an(/a?'s=---8ll, or xssy/\\. o In like manner^ if f- 10 = 20, then will 2j?2 = 60, an(/x«+ 15 ass 30, or j:2±=30— 15 = 15, or V^15. RULE VI. Any analogy or proportion may be converted in equation, by making the product of the two mean f equal to that of the two extremes. 7%M«, i/3x : 16 : : 5 2 10, then will Sx X 10 = 1( 80 8 or 3O:c=80, or x^ss — = — = 2|. ' 30 3 ^ 2 'P 2f ^ Andy if — :a:i b : C) then will — - sssab^ and 2cxs a li ■ Sab or xss ---. 2r 1 I I SIMPLE EQUATIONS. 79 In like manner jif 12— a? :— : : 4 : 1, /Ae« Wf// 12 — x^ ^x 12 ■raa2x, and 2x+j:=a:18| or jrs=---=s4. RULE VII* ^f the same quantity be found' on boAi sides of the ^QUation with the same sign, it may be taken away from ^fch ; and if every term in an equation be multiplied or <}ivided by the same quantityi it may be struck out of ^^em all. / TAt/», if 4x+aaa:*+a, then will 4xssb, and x= b ^f^ if 3oi:+5a*=B:8ac, then will Sx+56=8c, and "^ * 3 2jr 8 16 8 In like manner j if-^ ^ =— — --} then w/// 2x= 1 6, ^nd xssB. MISCELLANEOUS EXAMPLES. 1. Given Sx-^^\S^Bi2x+6 to find the value of or. Fir9tj 5a:— 3j:=b6+15, Orf 3x«=6+15«=21; SI ^ Jnd therefore xs«— =7. 2. Given 40— 6j7— 16=120 — 14j: to find x. Fintj 14j>-6j:=s120 — 40+16, Or 8a7»136— 40bs96 ; 96 And therefore j?«r— a«12. 8 80 SIMPLE EQUATIONS. > • 3. Let Sax — 3bs=i2da:+c be given to find x. Firat^ Sax — 2dx=iC+Zb, Or (5a — 2d)xx^c+3b. . c+3b And therefore 3?= 4. Let 3j:2 — \Ox^=Bx+x^. be given to find x. Firsts 3x — 10=8+x. Or 3x — x=8H-io=l8. • 18 And there/ore 2x=s\ 8^ or ar=--.s=9. Jt 5. Given ^ax^ — Xlabx^-sizZax^Jc^^^S to find x. Firsts dividing the whole by Zax^^ we shall have^ Or 2x — x=2+4b. Whence x=2 + 4b. %J^ ^^ %A^ ^^^k 6. Let r"+-r= ^^j ^^ given to find x. • 2 3 4 Firsts X 1 = 20. 3 4 Aho^ 3a— 2x-^ = 60, 4 ^nt/ 12a: — 8jr+6x=:240. Therefore 10j7=^40. ., ^ 240 And x==-7— =24. 10 ir-"~~3 «>c X"^^ 19 7. Given •— - +--=20 to find x. 2 ^3 2 2a i^e'r*?, X — 3H =40 — jr+19. 3 ^/«o, 3ar — 9+2a:=!20— 3JC+57. Therefore^ 3a:+2a+oa-= 120+57+9. 7%flrw, 8xxb|86, orx=— =23i. SIMPLE EQUATIONS. 8 1 2j? ^ 8. Let v^— +5aB:7} be given to finder. o FivBt^ ^— =7—5=2. ' ^ 3 Whence — =2^^^. 3 13 jind 8x=a 1 2, or ^r:^---= 6. 2 2aa 9. Let j?+v^fl»+xSaa . g , a be given to fiirdx. /Vr«f, J7^a »+xg+ a«+3:g=s2ag. Whence Xy/a^-^-x^^a^ — xS Or a^x'+x-^ssaa^ — ^2a«x»+x4, Or a*j7»-f-2a*j:2=sfl4, or3fl2j;a=a^; Consequently x^=s — ^ awrf x=!\/— =«%/—• EXAMPLES FOR PRACTICE. 1. Given 3y— 2+24=31 to find y. Jna. y=3.* 2. Given or+lSssSx — 5 to find x* Ana. ar=l 1^. 3. Given'6— 207+10=20— 3x^2 to find X. jin^, ars=2. 4. Given x+\x+^xcs\\ to find x. Ans, x=l§ 5. Given 2j:— 4x+l=5x— 2 to find x. ^/z«. x=-. ?• 6. Given 3«x+— - — 3=djr— 4i to find x. ^n9i x=-— -—•. 6a— 2<!» 82 SIMPLE EQUATIONS. 7. Given ^x+^x — Jx=b-^ to find x. Ana. x I 2 .*, 8. Given \^ 12-1- X =2+ v'J? to find jr. An%, x^ 9. Given j?+a= to determine jt. Ana ,xss — a+x 10. Given ^/a^+x^ssz(b^+x^)* to find x. Ans* j7M=v^ 2a 11. Given ^jc+^a-f-x-i to find x. 12. Given 4- =^, to find x. 13. Given a+x=v^a»-f-jc\/^+x^ to find a-. «i;{9. Xss — i— 4a PROBLEM I. 7 exterminate two unknown quantities^ or to reduce thet% simfile equations containing them to a single one, ^ RULE I. 1. Observe which of the unknown quantities is least involved) and find its value in each of the tions, by the methods already explained. 2. Let the two values thus found be made equal ead^ other, and there will arise a new equation only one unknown quantity in it, whose value xnajl found as before. . ♦•♦ SIM]|||^ EQUATIONS. 83 £Xi;||j|L£S.* 1. Given |^^+^^-^J^,tofind*andy. From, the first equation x= - 2 2 10+2y And from the second x= — — ; 5 23— 3y 10+ 2y Consequently ■■■ ■ ^ = > <« d Or 115 — 15y=»20+4y, Or 19y=115 — 20=95; 95 That isy y= --- = 5, ' ^ 19 ' 23 — 15 jfnd j7=— — =4. 2 2. Ghren ^ "^"^^^^ ? , to find x and f From the first equation jr=sa-— y, And from the second x^^b-\-y; Therefore ct^^y^ib-^y^ or 2y=ia — b ; a <o Consequently y=s , and x=a — y, _ a— -6 a-^b Or x^a = — — . ^ 2 . 5. Given ^t^tl^""!? , to find x and y. 2y From the first equation ar=:?14— ~-, 3 . • 3v And from the second Xiss2^^-^—f^*^' rr. 84 SIMPLE EQUA'qpNS. Therefore 14 — H£s=24— — , And 43 — ^^ys3iNi&— ^) Or 84— 4ysl44— 9y; Whence 5^=144— -84f^609 60 Andy^s^ — =12, ^ 5 ' 2y , 24 ^ 3 3 4. Given Atx+yss^^^ and 4^+^=169 to find x ai wfntf. ^ssBy and y ^ ^,. 2j: 3y 9 .Zx 2y 61 ^ 5. Given _ + _=_, and -+^ = 725. to fi. and y. ^ 1 j 2 ^ 6. Given x+ysssy andx*— y2=(/, to find ^ and y. i ^ r ^. 2* ^ S 7. Given X'^yssdy and ^ : y : : n : m^ to find x an RULE II. . 1. Consider which of the unknown quaflities ] would first exterminate, and let its value be found that equation where it is least involved. sNBubstiti^te the valub, thus found, for its equal the other equa^p, and there will arise a new equati with only oi^^lkknown quantitf, whose value may /bund a^b^brc. ^^y.^ SIMPLE EQUATIONS. <S SXAMPJ.E8. *• Giwn J 3^^j; • JJ., to find X and y. -^rom the first equation^ j?s=sl7-— 2y. •'^nd ^At« value substituted for x in the second^ gives - (17— 25/)x|Ji-y=«, Or 51 — 6y — V5s=2, or 51— -7y=2 j 7%a/ M, 7y =5 1 — 2=49 ; 49 Whence y=s — =7, anrf j?s5b17 — ^2y«17 — 14=3. 7 3. Given ?^^^^^3 ? > to find a? and y. jFVoto 4 he first equation jr=Bl3— j{. '^t' fAiff value being substituted f:>r x in the 2</y J»VM 13—1/ — y=:3, or 13 — 2ys:3, That is^ %= \ 3—3= 10, • »» 10 ' Whence ysa — =5,a72(/x=13— -y=13 — 5=S. '•Given J«;*^«;f;:^^, to find^^y. The first analogy turned into an equation^ oy ia bxssai/f or x=s -— , ^ <in£f /^ -va/ue qfx being subfttituted in the 2cf, Or ca|l+d3y2=fii2, or y^^-rrsi V 86 SIMPLE EQUATIONS. 4. Given 2x + 3y = 16, and 3a: — 2i/ = ll and t/. , jins* x=5j 5. Given — 5|- 7y = 99,4pid — + 7^7 = 5 1 and J/. •^/w. :r=7, ( 6. Given— —12=-^+8, an^^ + — - +27, to find 3c and j/. ^/;z5. j:=60,* 7. Given a : 6 : : x : y, and x^— t/S^s^, to fi nuLE in. 1. Let the given equations be multiplied or such numbers or quantities as will make the t contains one of the unknown quantities the s; equations. ^^ 2. Then, ^^fcding or subtracting the eq cording as is^requirtd, there will arise a ne with only one unknown quantity, as before. EXAMPLES* . Given p^JJ^^uJ.tofind^^andi Firhty viuUiply the 2d elation by 2 Vfill give 3 J? + 6t/ =42. Then from the last equation subtra and it ivitl give 6i/— 5y = 42 — 40 t ^nd therefore ;r sl4— 27/s:14-vi= n < SIMPLE EQUATIONS. %7 2. Given J 3^75 p ^^ , io dad X and y. -Let the first equation be multifilied by 2, and the 2d by '^i and we shall have Ifi^"^ 6z/=5^ ^fnd if the former of these be subtracted fro fn the latUr^ it will give iXysz&l^or ysB — =s2. fi 1 9 + 3y ^nd conieguentlyxsss — _, by the first equation^ 5 9 + 6 15 Ancftht'r method, ■Mukifily the first equation by 5, and the second by 3; and we shall have 5 ^"^"T, ^^^** ^ o ^o«, /rr /Ae*f rwo equations be added together^ 93 and the%um will be 31j|p:93, or ^ses-— a=3. 16 — 2x . mm And consequently^ y*= — — , by tXt^kond equation^ 16 — 6 10 ^ Oryss s — asi2, as before. MISCELLANEOUS EXAMPLES. 1. Give* ^^1^ + 8y==3l, and ^^ + 10x« 193, to 3 4 •lad JT and y. Ans. x=*\9j and ysn 3. 2. Given — h 14=18, and _I— +16«l9,to find X and y. ^«. •■sS, and y 88 SIMPLE EQUATIONS. . 3. Given — ~-i + ~ =:8, and -^-- — — j/=s find :r and z/. -^na, xa6, ancf 4. Given ax + byss r, and djp + eyss/j to find x ; r.^— */• . a/ jfna. xsB — -, ana yss — PROBLEM II. 2'o exterminate three unknown guantitiesy or to redu three aim/ile equations containing^ them to a single ;_ RULE. 1 . Let X, y, and z be the three unknown quantit be exterminated. 2. Find the value of x from each of the three equations. 3. Compardl|p fir&t value of x with the seconc an equation woRrise involving only y and z. 4. In like manner, compare the first value of x the third) and another equation will arise involving y and z. • 5. Find the values of y and z from these two equs according^ to the former rules, and Xy y, and z vv exterminated as required. ^ » M'ote, Any number of unknown quantities inay t terimnated in nearly the same manner; but ther often much shorter methods for performing the c tiooi which will be best ieamt from practice. rnXFLE EQVATtONS. t9 ^- Given -< jr-f2y-f 3za*62 J-, to fiiid j:, t/, wid z; i^r(W2 thejirst arssr 2 9— y—z. -fj'om Me second xssS^ — ^2y— 3z. ^ 2z/ z From the third a:=20— .-^^— w; 3 2 Whence 29 — i^— Z3a62--.3y— ^z, 2v z ^^flf 29 — y — z=:20— — . ^ 3 2 Also from thefirnt of these ys=:33.i— 2r. 3z And from the second yaB27— • — ^ Therefore 33 — 2zass27— — , or z^sil2r Whence also y=s33— -2z=9, Andx^29 — 9 — ^z=8. fi^+iy+i^«62-) Jt 3. Given -< -J^+iy+l^— 47 V, to fuH jr, y, and z. ^ir*r, the given equations^ cleared of fractions^ become 12x+ 8y+ 6z= 1488 20jc-f 15y+ 12z==2820 30jr+24y-f20z=4560 And^ if the second of these equations be subtracted from^ double the firsts and three times the tHrd from five Hmem, ^ht secondy we shall have 4x+ ys»l5« 10x+3yaB42a H2 oa SIMPLE JEQPATIOMS. And agatiij if the wond qf thewe be wbfrac^ed /r9i three ttmea thejfrat^it will give 48 fVr«/prtfy SB 156 -i-4x«: 60, (wrfr«sB — ■ g 3. Given jr+y+zss53,a:+2y+3z=a 105, and jr-f- -(-4zsB 134, to find X| y^ apd z. ^n«. a:s=24, vss6, ancf z=^ 4. Givea jr?+ycitf| ar+mft, and y+zaasc, to find y, andz. f. Given < dx+ey+/z9sn v, to find Xj y, and z. A COLLBCTION OF ;|UESTIOMS PaODUCIN« SIMPLE EqUATXOHS. 1. To find two numbers, such that their sum shall % 40, and their difference 16. Let X denote the least of the two numbevM reqiured; Then will x-^- 16 as the gr eater f And x-\-x+ I6xs40, by the question^ That M, 3xa40— 16s24, 24 Or jrsB— >aBl3«3/«<r«^ number ^ 3 And x+ 16eBl2+ 16n283»5r«a^^ number requf SIMPLE EQUATIONS. 91 2. What number is that whose ^ part exceeds its ^ part by 16? Lei x^Btnumbtr required; Then willits \fiart be —-, and tie ^ fiart — ; And therefore s=sl6, dy the question^ 3 4 That iSf x-^ — =48, or 4x-^3x^ 1 92 > XVhence xsas 192, the number required, 3. Divide lOOOl. between A, B, and C, so that A stisdl ^^\e 731 more than B, and C 1001. more than A. Let xsssh'a share qfthe given aum^ Then will x+72ssA'« »Aor^, j1ndx+l7^vaC*s3harey Mind the aum of all their aharea jc+-^+72+^+ 172i Or 307+244=1000, by the queation;. That ia^ 3:r=r 1000—344=756, 756 <9r a:=-l-=252/.=B'« ahare. 3 And x+72=252+72=384/.=A'* ahare. .^fiflrar+ 172=252+ 172=424/.=C*» ahare, 252/. 324/. 424/. i lOOOl, the firoof, 4. A prize of 10001, is to be divided between two per- sonsy whose shares therein are in the proportion of 7 to 9 ; required the share of each. Let x^tfirat fieraon'a ahare ^ Then will lOOO'^'mX ^maecond fieraon* a ahare, And X ; 1000 — x iiT i9^ by the queationf 92 SIMPLE EQUATIONS. That w, 9ar«.(1000— vc)x7«7000— rjF, Or 9x+7a7=16j:s=«7000, 7000 Whence x:=- ==43^/. \09,^Ut ehare^ 16 w^wdlOOO — :r=lOOO — 437/. 10«.=a563/. \Oa,^d share, 5. The paving of a square, at 2s. a yard, cost as much as the inclosing it at 6s. a yard: required the aide of the square. Let x=s8ide of the square sought^ Then 4x=yards ofindosufe^ And x^sszyards of fiavement ; Whence 4xX5ssi20x=z/irice of inclodngy jln d^x^ X 2 = 3^2 s=/2 rice of paving; But yx^ssaaOjT, by the question^ Therefore 2x:=i20,- and xi:x\0=: length of the side re- quired, 6. A labourer engaged to serve for .40 days upon these conditions, that for every day he worked he was to receive 20d. but for every day he played, or was absent, he was to forfeit 8d. ; now, at the end of the time he had to receive II. lis. 8d.; it is required to find how many days he worked, and how many he was idle. Let X be the number of days he worked. Then will 40 — x be the number of days he was idle^ Also xx20=:20j:=s«Mm earned,^ And (40 — jr)x8=S30 — ^xssisum forfeit edy Whence 20j7--(320— 8x)=380rf. (I/. 1 1«. 8c/.), by the question; that ia^ 20j? — 320+ SxsaSSO^ Or 2807=380+320=700, And ^==-r — =25=s7iMm^er of days he workvd^ And ^~^xaaa40<^26satlSjearnutnber of days he was idle. SIMPLE EQUATIONS. 93 Out of a cask of wine, which had leaked away |, 3 r >ns were drawni and then, being guaged, it appeared I half full ; how much did it hold ? Let it be aufifiosed to have held x gaUons^ X Then it would have leaked — gallona^ X conaeguently there had been taken away2\ -{ — galiona. X X But 2 1 H isz~^y by the question^ 3 i 3x That isf 63+xas — , Or \^6+2xssSx. Hence 3a:— .2x=l26, Or x:^ 126= number ofgaUona required^ What fraction is that, to the numerator of whichi be added, the value will be ^ ; but) if 1 be added to denominator, its value will be ^ ? X Let the fraction be refireaented by ^» ; V Then vnll —I— = — , ^nd = — , y+{ 4' Or 3x-|-3=y, ^nd 4x«=y+l, Hence 4x— 3 j:— 3 = y + 1 ---y, Thatia^ x — 3=1, Or ar=s4, and yBB3x+3-il3+3=3l5, 4 So that — ^ss fraction required. 94 SIMPLE EQUATIONS. 9. A market-woman beught in a certain nnmber eggs at 2 a penny, and as many at 3 a penny, and s them all out again at the rate of 5 for two-pence, a by so doing, lost 4d. what number of cgjjs had she ? I^et xss number qfeggs of each nort^ Then will -^^szfirice (ifthcjirat aorty X And — as hrice of the second sort ; 3 . But 5 : 2 : : 3x {the whole number of eggs) : • 4uC Whence — firice of both aortsy at Jive /or 2d% 5 And 1- ^— — — a»4, by the question ; 40 «i d ^ . %x %x That i«, x-\ ==8 ; 3 5 24j7 Or Sx+2x a«24; Or 15x+10ar — 24^=120; Whence xs=s 120= number of eggs of each sort require 10. If A can do a piece of work alone in ten days, a B in thirteen ; set them both about it together, in wl time will it be finished ? Let the time sought be denoted by jt, X Then 10 days : 1 work : : x days : — , X And 13 days : 1 work : : xdays : •— ; X Hence — tssftart done by A in x daysy X And — ^sfiart done by B in x days; 1 o '^S' SIMPLE EQUATIONS. 95 X Consequently — H s=: 1; 13jr , ' Thatis^ h xs= 13, or 13x+10:r= 130: 10 . 130 And therffore 23x3sl30, or jc = -— = 5|^ daya^ the time required, 11. If one agent, A« alone, can produce an effect e, in (he time a, and ariother a^cnt^ B. alone, in the time b ; in what time will they both together produce the same effect? Let the time nought be denoted by x. ex Then ai e i i x i — =^iiart of the effect firoduced by A, tt ex .-M, bi e I I X I ^^ fiart of the effect firoduced by B; ex t'^ Whence 1 =: e^ by the question ; a ^ ^ X X Or--4--=i; a b That i«, xA =a ; b Or bx-^ax^ssba ; And consequently x=. == time required. Ql^ESTIONs FOR PRACTICE. *• What two numbf:rs art those, whose difference is 7. ^'^^l sum 33 ? Ans. 1 3 and 20. ^» ^« To (livide the number 75 into two such parts, that • 5^e times the greater may exceed seven times the less ^* 15. Ans» 51 and 21, 96 SIMPLE EQUATIONS. 3. In a mixture of wine and cfder* ^ of the whole /i/im 25 gallons was wine, and ^ part minua 5 gallons was cjr-^ der ; how many gallons were there of each f jirm. 85 qfwiney and 35 of cyder ^ 4. A bill of 1201. was paid in guineas and moidore^. and the number of pieces of both sorts that were use^^ was just 100; how many were there of each? Jna, 50 ofeac^^ 5. Two travellers set out at the same time from Lon- don and York, whose distance is 150 miles; one of then goes 8 miles a day, and the other 7 ; in what time wiU they meet ? ^na, in 10 dayv^ 6. At a certain election, 375 persons voted, and th^ candidate chosen had a majority ol 9 1 ; how many voted for each? Ana. 233/or une^ and \42/br the othewr'^ 7. What number is that from which, if 5 be rot>* tracted, f of the remainder will be 40 ? jina. 6S« 8. A post is \ in the mud, ^ in the water, and 10 feet above the water \ what is its whole length ? jina* 24t Jeet, 9. There is a fish whose tail weighs 91b. his head weighs as much as his tail and half his body^ and'his body' weighs as much as 'his head and his tail ; what is the whole weight of the fish? Ana.T^it. I| 10. After paying away i and | of my money, I tad 66 |;uineas left in my purse ; what was in it at first ? Ana* ]20 guineai* 11. A's age is double of B% and B's is triple of C% and the sum of all their ages is 140; what is the age of each? Ana. A'*=«=84, B'«=42, and C'4««i4 12. Two persons, A and B, lay out equal sunu of money in trade ; A gains 1261. and B loses b71. and A'a money is now double of B's; what did each lay out? Ana. 3Mi I > SIMPLE EQUATIONS.'^ 97 13. A person bought a chaise, horsei and harness, for ^. the horse came to twice the price of the harness, and ^'le chaise to twice the price of the horse and harness ; ^hat did he give for each ? jins, 131,68, 8d,Jbr the horse^ 61, ISa, Aid, for the harness^ and 40i,/br the chaiae, 14. Two persons, A and B, have both the same income : ^ saves -J of his yearly, but B, by spending 501. fier an- i ^Um more than A, at the end of 4 years finds himself 'OOl in debt: what is their income ? »/ina, 125/. 15. A person has two horses, and a saddle worth 501. ^f)W if the saddle be put on the back of the first horse, it . *^11 make his value double that of the s&cond ; but if it "^ put on the back of the second, it will make bis value Uiple that of the first ; what is the value of each horse ? ^w«. One 301, and the other 40/. 16. To divide the number 36 into three such parts that •} of the first, \ of the second, and ^ of the third, may be ail equal to each other, ,/^na. The fiarta are Qy l^^and i6. 17. A footman agreed to serve his master for 81. a year and a livery, but was turned away at the end of 7 months, and received only 21. ISs. 4d. and his livery ; what was itH value ? >!fna, 4/« 16^. 18. A person was desirous of giving 3d. apiece to some beggars, but found he had not money enough in his poc- ket by 8d. be therefore gave them each 2d. and had then 3d. remaining ; required the number of beggars, j^na, 1 1 . 19. A hare is 50 leaps before a greyhound, and takes 4 leaps to the grcyliound's 3 ; but two of the greyhound's leaps are as much as three of the hare*s ; how many leaps must the greyhound take to catch the hare ? Jna, 300. 98 SIMPLE EQUATIONS. 20. A person in play lost A of his money, and then woi r> shillings ; after which he lost | of what he then ha( and then won 2 shillings; lastly he lost ^ of what he thei had; and, this done, found he had but 12s. remaining what had he at first ? jins, 20a 21. To divide the number 9CV into four such parts that if the first be increased by 2, the second diminished b] 2, the third multiplied by 2, and the fourth divided by 2* the sum, difference, product, and quotient shall be a] equal to each other. jlna. The parts are 18, 22, 10, and 40, reafiectivel^ 22. The hour and minute hand of a clock are exactT together at 12 o'clock; when are they next together? jlna. I ho. 5r^ nti S3. A man and his wife usually drank out a cask < beer in 12 days; but when the man was from home, lasted the woman 30 days ; how many days would the mjE alone be in drinking it f .^ns, 20 dai^ 24. If A and B together can perform a piece of wor in 8 days ; A and C together in 9 days ; and B and i in 10 ^ys'f how many days will it take each person t< perform the same work alone ? Jna. A 14-11 daysj B iT^^and C 23/j. 25. If three agents, A, B, and C, can produce the ef- fects a, A, c, in the times e, /, g^ respectively ; in ivhat time would they jointly pix)duce the effect d? V [ 99 ] QUADRATIC EQUATIONS. •^ sim/ile quadratic tquation is that which involves the *^Uare of the unki>own qiiaiitity only. •An adjected quadratic equation is that which involves tJie square of the unl;.nown quantity, together with the product that arises from multiplying it by some known quantity. T'huB ax^ = 3, is a simfile quadratic equation^ -^nd ax^-j- bx = c, is an adfected quadratic equation. The rule for a simple quadratic equation has been given already. ^ All adfected quadratic equations fall under the three '^Mowing forms : 1. a72«j_ axsszb 2. x^ — ax=s b 3. x'^-—^ax^=i-^b. The rule for finding the value of x^ in each of these ^^uatioDs, is as follows: RULE*. 1. Transpose all the terms which involve the unknown quantity to one side of the equation, and the known terms to the other, and let them be ranged according to their di- mensions. • The square root of iny quantity may be either-j-or — , and there- fore all quadratic equations admit of two solutions. Thus the square root of -f-'i^is -j-M or — n; for (+«) X (+'^) o' ( — ") X ( — ^) arc each equal to -{-«2, but the square root of — «2 or ^ — «2, is ima- ginary or impossible. ;v,v,vAv;\\ 100 QUADRATIC EQUATIONS. 2. When the square of the unknown quantity has an; co-efficient prefixed to it, let all the rest of the terms b divided by that co-efficient. 3. Add the square of half the co- efficient of the secon< term to both sides of the equatioUi and that side whicl involves the unknown quantity will then be a complet square* 4. Extract the square root of both sides of the cqua tion, and tlie value of the unknown quantity will be de termined, as was required. J^ote 1. The square root of the first side of the equa tion is always equal to the unknown quantity, with ha] the co-efficient of the second term subjoined to it • 2. All equations, in which there are two tenms in volving the unknown quantity, and which have the inde: of the one just double that of the other, are solved, lik quadratics, by completing the square. Thus^ x^ + 0x2= d, or x^^ + ax«== d, are the' same a quadratics^ and thf, value of the unknown Quantity may b determintd accordingly. since either of them, being multiphed by itself, will produce ^-f-^oS And this ambiguity is expressed by writing the uncertain sign J before y/ib+ia^)-, thus Jf=±v'(^+ia»)-- i^- In this form, where x:am±.^(b-\-la^) — \aj the first value o X, viz. x=-f.^(A — \a^) — -Ja is always affirmative; for sine ^€fi-\-b is greater than ^^2, the greatest square must necessarily hav^ the greatest square root; therefore ^/(^b-\-^a^) will always be great er than v^ia^, or its equal la; and consequently -}-^(5_j_^ii2^ — ic will always be affirmative. I QUADRATIC EQUATIONS. 101 EXA.MPLES. 1. Given j72+4x=i40, to find x. first J ar3+4x+4= 140+ 4=144, by completing the *9Uare, Then v^(j?«+4j?+4) = v^144, by extracting the root; Or, which is the same thing, :c+2=s 12, ^ JInd therefore x^=\2'-^2^=^\0. ^. Given x^ — 6jf+8=80, to find x. First, x^ — 6^7=80 — 8=72, by transposition ^ /^en x^ — 6^+9=72+9=81, by completing the square, Jlnd JT— 3=-v/81=9, by extracting the root ; Therefore x=9 + 3=l2. Tbe second value, viz.x=s— y/(^+^a*) — ^a, will always be^ne- itive, because it is composed of two negative terms. Therefore, "^Vhen x2+ax=A, we shall have x=+^(A+da3) — ic for the Affirmative value of x, and xz=s. — \/{b-\-~a^) — la for the negative "Value of X. In the second form, where x=s-^y/(b-^la^)-{-iay the first value, Viz. x=+y/(^+ a2)-|_^a, is always affirmative, since it is com- '^osed of two affirmative terms. The second value, viz. x= — y/(b ^^a2)+ia will always be negative; for, since A+i^a is greater than ^a», \/{b-\--^a^) will be greater than ^-fl2, or its equal ^a; ^nd consequently — \/(^~t--l^^)+i^ ^^ always a negative quantity. Therefore, when x2 — a.v=^, we shall have x-=z-\.^/(b-\-^.a^)-^ -Jfl for the affirmative value of x, andx=: — ^/(J>-\- a'i)-^.a for the negative value of x; so that, in boih the Hrsv und second forms, the unknown quantity has always two values, one of which is positive^ and the other negative. 1 2 i -^if lOS QUADRATIC EQUATIONS. 3. Given 2a:2+8j? — 20=70, to find x. ' Firstj 2x24-8ar=70+20=90, 6y transfiosUion. "'! L Then x^+4x=s^5y by dividing by 2, jind :r2+4x+4=495 by comfileting the square ; Whence x+2:s:iy/4i9^=7^ by extracting the root, ^nd consequently x==7 — 2=5. 4. Given 3x^ — 3.r+6=5^, to find x, "^ Here x^ — :r+2 = l^, by dividing by 3, ^nd x2— a:ss: 1^ — 2, by transjiosltion ; Also x^ — jc+^= l7..«.2-f. 1 -s-^^j by comfileting the squc And X — J=^yig-=J, by evolution ; Therefore j:=J-|-^=7) the answer. 5. Givfen ^-— ^ +20A=42|, to find x. x^ X Here --— — -=42| — 20A=22|., by transfiosition^ 2x And x2— — =44.|, by muUi/ilying by 2. Whence x^ f-i=44^4-^=44|, by comfileting squarcy And X'~^ssiy/4f4^-^6^y by evolution ; There/ore a; was 6^ -4-^=7, the answer^ In the third form, where x=sdts/(icfi — ^)-j-Ja, both the vs of X VfiW be positive, supposing la^ h greater than b. For the "'. vvalue, viz. x=-f-^(ifl2 — 6)+^a will then be affirmative, b composed of two affirmative terms. The second value, viz. x= — y/Qa^ — b)-\-ia is affirmative! since ^a^ is greater than r^fl*— J, y/^fl*, or la, is greater \/(id^ — b)i and consequently — ^(^a2_«^)«|_jflf will. always b affirmative quantity. Therefore, when x^ — axssz — b, we shall ! .v=+v^(ia3— Z»)4.^cf, and also x=— v^(ia2-H^)+ffl, for affirmative values of x. ^ QUADRATIC EQUATIONS. 10 A ^» Given flx*+dx=:c, to find x> r» be ^irst^ x^-^ xs= — , by division; a a ^ b b^ c b^ rn, x^-\ j7 -J ss 1 by comfileting the square. a 4a* a 4a* « ^ b f c b^\ /4ac+d2\ , • Given ax^ — te+c=rf, to find x. •Herey ax^~~-'^xs=sd.,^^Cf by trans/iositioriy Jlnd x2— — x= , by division, - a a b b^ c?— c b^ ^ x^ X'\ =— 1 ^by comfileting the square^ a 4a2 a 4a^ b Id — c b^ \ And ^--±V ^-^+ ^,j, by evolution ; Therefore x^ ±±y/ f — +— V ^. Given x^+2ax^sstby to find x. ^ej x^-i-2ax^'\'a^=::b+a^y by comfileting the square^ And x2+.a=s^(d4.a2), by evolution ; Whence :c2«v^(<5+a2>--a^ " And consequently j?=V — a+y/^b-^-a^). But in this third form, if ^ be greater than Jfl«, the solution of i proposed question will be impossible. For since the square of any intity(whether that quantity be a^rmative or negative) is always Tmative, the square root of a negative quantity is impossible, and mot be assigned. But if ^ be greater than ^fl», then -40% — b is a (ative quantity; and therefore ^(JfC^ — b) is impossible, or ima- lary ; consequently \tk that case xssiiet±iy/{\a^^b) is always possible, or imaginary. 104 QUADRATIC EQUATIONS. n 9. Given aj?" — bx^ — c=— </, to find x, n Firsts ax^ — bx^^sszc—^^ by iranafioaitiorij b n c d jind x^ x^=s , by division^ a a d n 1,2 C-— e/ b^ JUo x" ;c^H = f-— T, by com/ileting t square, n I, /c— ^ b^\ , And x^ xsik/ I h — :; ) by evolution ; n b /C — d b^ \ ^wa consequently x= [ — ±v^^ ^ J n • EXAMPLES FOR PRACTICE. J. Given x2 — 8jr+10=sl9, to find jc. Ans,x=^ 2. Given x^-^-x — 40=170, to find x. Ana, x=l 3. Given Sx^+Sx — 9=76, to find x. Ana. x=^ 4. Given ^x^ — Jx+7|=8, to find x» Ana, x=s i 6. Given 2x^ — x2=496, to find x. Ana. xsss 6. Given 4^ — Jv^x=22|, to find or. «^n«. ^=4! 7. Given |ji;2+3^==|, to find x. ^n«. .6685 8. Given 07^+6x3=2, to find j:. . Ana. x^^—S+s/U 9.' Given x^+x=ay to find x. Ans.xs=\/(^a+\) — i 10. Given j?— v^jr=o, to find x. _ Ana. x=(i+V^«+iJ 1 1 . Given 3x^n — 2*'»=s25, to find x. -^na. :r=(^v^76+^ 12. Given \/\+x — 2^^l.fj?=4, to find x. Ana. x=Cl+-v/5>— QUADRATIC EQUATIONS. 105 f QUESTIONS PRODUCING QUADRATIC EQUATIONS. I. To find two numbers whose difference is 8, and product 240. • Let x=Bthe leas number i Then will x+Sssthe great er^ ^nd xXi^+S)=x^+Sxss24iOy by the question; Whence j??+8j74- 16=240+ 16=256, by completing the square, jiUo jr4-4=\/256=16, by evolution; '^nd therefore j:=16 — 4=:12aB/f5« number ^ and 12 + ^^20ss greater* 3* To divide the number 60 into two such parts that their product may be 864. Let x=sgr eater fiart^ Then will 60— x=/eM, 4nd xX(60— a.)=60j:— J72--864, by the question; That M, x^ — 60j:= — 864 ; ^^hence or*— 60x+900= — 864+900=36, by completing the square. Also X — 30=v'36=6, by extracting the root; Jnd therfifovi? x=6+S0=x36^ greater fiart) And 60 — a?=60 — 36=24 = /<?««. 3. Given the sum of two numbers =10 (a), and the sum of their squares = 58 (6), to find those numbers. Let Xiasgrjater of the two numbers; Then will a— x= less; And jr*+(a— x)2=2a'2+aa — 2cix=d, by the question^ a* b Or x*+ — — ax^ — J by division. 106 QUADRATIC EQUATIONS. b a^ Ih'^a^ Or X*— cj?=» — — — =ssr. , by iransfiosi a2 ^«_a2 q2 2 b — a^ IVkence x^— ^cj: +-^ = — 7: 1 = > ^y 4 2 4 4 ing the square, a 2b-'-~a^ Mao X s= ^ , by extracting the <6 4 2b a2 a And therefore a?=±v/ 1 ^= greater n . . (a ^2d— an ^2d— fl2 a Hence these two theorems^ being fiut into numi 7 and 3 /or the numbers required* 4. Sold a piece of cloth for 241. and gained fier cent, as the cloth cost me ; what was the pr cloth ? JLet xsssfiounda the cloth coat; Then 24i^-x=swhole gain; • £ut 100 : X I : X : 24— a:, dy the que Or x2=100X(24 — x)=2400— IOOjt; That ia x^+ 100ar=2400 ; Whence x^+ iOOa:+2500=2400+2500=490C fileting the aguare. And j:'+50=:-v/4900==70, by extraction of n Consequently xssx7O--~^50=:20l=:/irice of the 5. A person bought a number of oxen for if he had bought four more for the same n would have paid U. less for each; how mai buy? Let the number of oxen be refireaented I 80 Then will — be the fir ice of each. X QUADRATIC EQUATIONS. 107 80 -^nd —r:=^f^^^^ ofeachy t/s:+4 had coat 80/. ^ 80 80 ^ut — = — —- +lj by the question* ^ 80x Or 80= \-x. x+4^ * Or 80x+320=80j7+a:*+4x, ••^ce ara + 4j;+4= 320 + 4 = 324, by comJUeting tife ^quaref And j?4-2=\/324s=b18, by evolution; ^^equently x=i 18 — ^2=16 == number of oxen required. r 6. What two numbers are those whose sum, produ(^, 'ana difference of their squares are all equal to each other ? Let a:= greater number^ And yzszleaa; And 1 = — --— ssx — y, or arx=y+ Ij/rom the 2d equation; x-f-y Also (y+ l)+y=(y+ l)Xyj from thefirat equ^iion^ Or 2y+l=sy2-^lyj 7%a^w,9«-y=l; Whence y*— y+^^l^j ^y completing the aquare; 1 5 %^5 ^/so y— — =^1^=:^ — s — -, by evolution i Conaequently y = -— H = -— — , and j?=s y + 1 ass ^- — 3I_. And if these exfireaaiona be turned into nuwbensy %ve ahall have a:s=2.6180+ fl7Zrfl/5rsl.6l80 + 108 QUADRATIC EQUATIONS. 7. There are four numbers in arithmetical progr^ sion, of which the product of the two extremes is 45| that of the means 77; what are the numbers? JL,et x^s^lesB extreme, and y s=scommon difference; Then x^x+yj x+2yj :r+3y, will be the four nimbert «"" ^(x+y)x(a7+2y)=^*+3ary+2ya=775 gue9H(M^^ 32 Whence 2y2--s 77—45 =32, by subtraction^ and y^Ga-^^s^ 16, by division, * Or yss^l6=s4, by evolution; _ Therefore x2+3xy«»:jp*+l2ar=45, by the Xst equation; " Mao 072+12^7+36=55 45+36=81, by completing the^ square^ And j:+6=^81=s9, hy the extraction of roots; Consequently x=9— *6s=3, and the numbers are 3, 7, llj^ and 15. 8. To find three numbers in geometrical progression^ whose sum shall be 14, and the sum of their squares^ 84. Let X, T/, and z be the numbers sought; Then xz=sy^y by the nature of firo/iortitmj •^'^^ \ ^2+y2+23^ Jl \ > *y '^^ quesHon; But j7+z=14— y, by the 2d equation* Jnd ar2+2j7Z+z2= 1 96— 28y +^2^ by squaring both sidei^ Or x2+22+2i/»=196— 28y+y2, byfiutting^y^.for itff equal 2xz; That is^ x2+z«+y2=196— 28y, by subtraction^ Or 196— 28y=84, by equality; 196 — 84 Hence yss _ =s4, by transfiositton and division. / QUADRATIC EQUATIONS. 4^9 16 fgairij a:zs=y2^16, or ar== — by (he \at equaiion, z 16 '^ndx-^-y-^-z^n ^- 4 + z = 14 by the 2d equation^ z « r^r 16+42 + z2=r 14Z, or Z^ lOZras — 16, IVhence z^ — lOz +25= 25 — 16 = 9, by completing the iguare^ ^nd 2 — 5 = v^9=3, orz = 3 + 5=8, Consequently^ x = 14 — y — z = 14 — 4 — 8 ==2, and the numbers are 2, 4, 8. 9. The sum («) and the product {fi) of any two num- ^^^rs being given; to find the sum of the squares, cubes, ^^cjuadrates, &c. of those numbers* Let the two numben be denoted by x andy. Then will i "^ •" ""* *> by the question. But (j:+y)*=x*+2jpy+y*=«*, by involution^ and x^'{'2xy'\-y^''^2xy=s8^ — 2/2, by subtraction. That is^ j7«-f-y25— «2 — 2fi9=i8nm o/ the squares. -^gainj (a:2+y2)X(a?+t/)=(«2— 2/2)X«, by multiJiUcation^ Or J734-j:yx(x+y)+y3=fi3 — 2*/i, Or x3^«^+y33--«3 — 2»/», by substituting .72 for its equal 3cyx{x+y)\ ./^^ And therefore x^+t/SssaS — lJ^/iaBssttm q/*/^? cubes* -in tike manner^ (x3-fy3)x(j?+i/)=(«3 — 3sfi)xsy by mul" ti/iUcation^ Or ar*+ary X(j?2+y2)+y4=:«4— 3«Vi, Or x^+/iX(«* — 3/i)+y*=«'* — 3*% by substituting fix, {«« — 2fi) for its equal xyx{x^+ y*) i Andt conset/uentlyy x^+y* aesi*— 3«V*'— /^ X («2 — 2/j) »=*<— 4s^fiJ^2fi^ ssi sum of the biquadrates^ or fourth powers ; And the sum of the nth, powers is 4" — ni^^-^p -f x . «— 3 ♦»•— 4 , « — 4 n — 5 w — 6^, , n — 5 n— .6 n — 7 w — 8 1 J-.* M^c. ^ no QUADRATIC EQUATIONS. to. The sum (a) and the sum of the squares (6) four numbers in geometrical progression being give to find those numl^rs. ^ Let X and y denote the two means. Then will — and ^— 6e the tivo extremes^ by the nati y X offirofiorlion. MsOy let the sum of the two means = «, and their fi duct =fi^ Then will the sum of the two extremes = a — fS by question^ and their product =s/ij by the nature qffi portion, fx^ + y^ ^ %., ij^ ^ 1 . Jitnce} x^ y^ ^^^ ^y Uy the iastfirobla Jnd x^ +y^ + ^ +^-«* + (a~«)2— 4/i=6, by y x^ question. x^ y^ Again^ 1 = c— », by the question, y X Orx3+i/=J?t/X(a— «)=//X(fl— «)• But j:^-4-J/^==*' — 3A/i, by the last fir oblem; And therefore fiX{a — *r)=A3 — 3«/?, by equality^ Qr pa — pH+:itis^pa-\''Zps^=^is^y Or p=s by division; - a+2» Whence «2+(a— «)2— 4/i=&a+Ca—«)2— — — =i, substitution, "^ b a^-^b Or s^"\ »= f by reduction, a 2 /a* — b ** \ * 1 and extracting the rooi, Andffrtm this value qftj aU the rest of the quantities Xy mnd y^^inay be readily determined. QUADRATIC EQUATIONS. 1 1 1 QUESTIONS FOR PRACTICE. 1. What two numbers are those whose si^tn is 20, and *^>eir product 36? Ans, 2 and \%, 2. To divide the number 60 into two such parts, that *^eir product may be to the sum of their squares in. the ^atio of 2 to 5. Ana. 20 and 40. 3. The difference of two numbers is 3, and the diffe- ^nce of their cubes i^ 1 1 7 ; what are those numbers ? 4. Ana. 2 and 5. 4. A company at a tav^nr hia^I. 153. to pay for their reckoning ; but, before fc'bU^ivii settled, two of them left the room, and then llrtirii'rwrtwjri iinini il had 10s. a- piece more to pay than JWIblBif iiow many were there in company ? ^ ^"^ Ana, 7. 5. A grazier bought as many sheep as cost him 601. and, afiei; reserving 15 out of the number, he sold the remainder for 541. and gained 2s. a head by them; how many sheep did he buy ? Am, 75, 6. There are two numbers whose difference is 15, and half their product is equal to the cube of the lesser number; what are those numbers? Ans, 3 and 18. 7. A person bought cloth for 331. 15s. which he sold again at 21. 8s. fier piece, and gained by the bargain as much as one piece cost him; required the number. of pieces. Ans. 15. 8. What number is that, whicn, when divided by the product of its two di{>;it3, the quotient is 3; and, if 18 be added to it, the digits will be inverted? Ans, 24. 9. What two numbers are those whose sum, multi- plied by the greater, is equal to 77; and whose diffe- rencC) multiplied by the lesser, is equal to 13 ? Ans, 4 and 7, or f^/S amd V \/3 >4r' % 1 12 QUADRATIC EQUATIONS. 10. To find a number such that if you subtract it from 10, and multiply the remainder by the number itself, the product shall be 21. ^na, 7 or 3. 11. To divide 100 into two such parts, that the sum of their square roots may be 14. jlns, 64 and 36. 12. It is required to divide the number 24 into two such parts, that their product may be equal to 35 times their difference. -^ns, 10 and 14. /■-> 13. The sum of t^o numbers ia 8, and the sum' of their cubes is 152 j what are the numbers ? ^ns, 3 and 5. 14. The sum of two numbers is 7, and the sum of their 4th powers is 641$ vhat tfre the numbers? .' ' Ma, 2 and 5. 15. The sum of two numbers is 6, and the sum of their 5th powers is 1056; what are the numbers? jina. 2 and 4. 16. The sum of four numbers in arithmetical pro- gression is 56, and the sum of their squares is 864; what are the numbers? ^na, 8, 12, 16, and 20. 17. To find four numbers in geometrical progression whcsc sum is 15, and the sum of their squares 85. ylua. 1, 2, 4, and 8. 18. It is required to find four numbers in arithmetical progression, such that their common difference may be 4, and their continued product 176985. ^ jina. 15, 19, 23, and 27. 19. Two partners, A and B, gained 1401. by trade; A's money vvas three months in trade, and his gain was GOl. less than. his stock; and B's money, which was 501. more than A's, was in trade five months;' what was A*s stock? ji7i8. 100|. % * [ 113 3 •F THE NATURE AND FORMATION OF EQUATIONS IN GENERAL, ■ All equations of superior orders are generated by the moltiplication of those of inferior orders, involving the same tinknown quantity. Thus, a quadratic equation is formed by the inuUipll'- cation of two simple equationt. A cubic equation is produced by the continued multi- plication of three simple equatioMf or by one quadratic and one simple equation. A biquadratic equation is generated by the continued multiplication of four simple equations ; or by two qua- dratic equations ; or by one cubic and one simple equa* tioii) See. For, suppose the unknown quantity to be x, and its values in any simple equation to be a, ^, r, </, &c. Then those simple equations^ by bringing all the terms to one side, will become x-— aaeso, x-— d^o, x — c ^ 0, X— d =s 0, 8cc, And the product of any two of these, as (x-~a) x (x— 6)b8so, will form a quadratic equation j or one of two dimensions. The product of any three of them, as (x — c) x (x— 6) X (x— c) = 0, will form a cubic equation^ or one •f three dimensions. r The product of any four of them, as (x — n) x (x— b) X (x— - c) X (x— >(/) ass 0, will form a biquadratic egua- tion^ or one of four dimensions, &c. From hence it appears, that every equation has as many roets as it has simple equations which produce^ I I i i K ATURE OP EQUATIONS. it, or as there are units in the highest dimension of the unknown quantity. For, if any of the values of x (a, b, c, d) be sub- stituted in the place of x in the biquadratic equation (r — a) X (x— 6) X (jc — c) X (x— rf), all the terms of. that equation will vanish, and the whole will be equal to nothing. And, as there are no other quantities, besides these four, which, substituted in the place of x, will n\ake the |>roduct vanish, it is plain tliat the equation cannot pos- sibly have more than four roots, or admit of more than four solutions. After the same manner, it may also be shown that no equation whatever can have looie roots than it contains dimensions of the unknown quaolity. To make this still plainer by an example in numbers ; suppose the equation to be resolved be x^ — IOjt^ -|- 35jp2_50j7 + 24 = 0, and that you discover this equ«» tion to be the same with the product of (x— -l) x (x— 2) X(x— 3)X(:c — 4). Then it may be inferred, that the four values of x are 1, 2, 3, and 4; for any of these numbers being put for X will make that product, and consequently x^ — \0x^ + 35j?2_50x + 24, equal to nothing, as it is in the pro- poaed equation. * And it is certain that there can be no other values of X besides these four ; since, if any other number bt; sub- stituted for X in those factors, none of them will vaaish, $fid therefore their product cannot bo equal to nothing, as it ought to be by the equation. The roots of ec^tions are, also, either fioeiUve or vegatrvey ftccording as the roots of the sin>p]e eqtiatiofi from whence they are produced are positive or negative. Thud, if you suppose a: =— a, xsxsdy xms — c, and xaa df Uien will x+avmoj x^^d^so^ x+Cf^Q^ wad 47-— d > , -i. NATURE OF EQUATIONS. 115 =0, and Ihe equation (x+a) x (jc— ^) X (^+c)x{x-^d) BO9 will have its r«ots — a, +dy — c, +d» But the »igne and co-efficienia of equations will b^ best inderstood by consideiMiig the following table; where he: si lYiple equations X — a, x — 6, Sec. being multipiitd UMiiinuaily togtiber, produce, successively, the higher :qualions. X*— ax X X— ^=0 — d Ix^-j-^^ VX— tf^CassO, a cubic, Xj'^-^ssO • — ar\ abc^ bd\^x '{' abcdssz 0, a biguadna* >x^ ^■^acd \ tlCy bed) &c. l^^rom the inspection of these equations it is plains tHat ^lie co-efficient of the first terms is unity. The co-efficient of the second term is the 9um of dU ^ht TQfkU (a, A, c, rf) vfith contrary eigns* The coefficient of the third term is egual to the sum ^tfihe reetangiea qfthe roots^ or of\>U the firoducta that ^«n Jioaaibly ame by combriirff them tvfo and two. lie NATURE OF EQUATIONS. The co-efficient of the fourth term is equal to t of all the firoducts that can /losaibly arise by con them^ three by three ; and so on for any other co-e whatever. The last term is always equal to the firoduct oj roots with contrary signs; and this reasoning wi "Whether the roots be positive or negative. It likewise appears, from inspection, that the s all the terms of any equation in the table are alte: -f and — . Thus the first term is always some pure powe: an^is positive. The second term is some power of x, multipl the quantities — 'a* -— ^, — r, &c. and, since these titles are all negative, it follows that the term itsel be negative also. The third term has the product of any two of quantities ( — a, — d, — c) for its co-efficient^ J therefore positive ; since — x — , as well as + X +j -f-, or an affirmative quantity. For the same reason, the next co-efficient, wh formed of the products of any three of these ne quantities, must be negative ; and the next fbllowin ing made up of the products of any four of the sa ^ative quantities, must be positive, and so on. And, from this reasoning, it plainly appears, that all the roots are fiositive^ the signs are filus and alternately. But if the roots be all negative, as jre=-— a, x^ 3?=— c, xss— ^i then (x+a)x(x+iJ)X(jr+c)X(. aso, will express the equation to be produced; a the terms will plainly be positive. So that, when all the roots of an equation are neg M is plain that there can be no change in the signs ( terms 9f that equation. And, in general, there will be as many pc roots ia vxkj equation as there are changes ii flSC ?/- 121' NATURE OF EQUATIONS. 117 signs of the -terms of that equation, from + to — , or rifl from— to +; and all the rest of the roots will be ne- L^l gative. From thb rule it follows, that, in quadratic equations, the two roots may be either both positive, or both nega* livC) or one negative and one positive. ThuS) in the equation, a ^ , + c^ = o, or (x — o)x Xj>— *), there are two changes of the signs, and therefore the roots are both positive. In the equation x^ J_ ^^ + ab^o^ or (j:+c)X(x+^), thei^ is ho change of the signs, and consequently they arc both negative. And, m the equation ^^ ■ ^ — 0^=0, or (x-— a) X C*+^)> otie of the roots will be affirmative and one ne- gative; for, as the first term is positive and the last ne- gative, there can be but one change in the signs, whe- \ Uicr the second term be -f or — . \ In cubic equations, the roots may be all positive, or all negative ; or two of them may be negative and one posi- tive, or one negative and two positive. Thus, in the equation (.r — a)y.{x — 6)x0r — r ) = 0, the signs will be alternately + and — ; and, as the number of changes is three, the roots must be all Ijositive. In the equation (or -f: a) X (J^ + *) X (j:* + c)=o, where there are no changes of the iigns, the roots must be all negative. In the equation :r2_(fl-_id4- c).a2 + (06 — ac — be) x 4. abc = 0, or {^x^^a) X (r — A) x (^^ -f- t-he number of changes will be two, and conseqneiiiiy two of the roots Avill be positive and one negative. For if + ^ be greater than c, the second .term must be negative, its co-eflicient being — a, — 6, + ^ ; 118 RESOLUTION OF EQUATIONS. and if a + A be less than c, the third term must be gative, its co-efficient + ab — ac — be being in that c negative. In the equation x^ -\' (a -{-b — c) x^ -f- («^ — «c — X'-^abc =ss Oj there can be only one change of the sig and therefore one of the roots is positive, and the otl tVvo negative. For if a + d be less than c, then the second terra negative, and the third must be negative also: and a + ^ be greater than c, the second term will be positi and there can be but one change in the other two ten whatever may be their signs. And, in the same manner, this reasoning may be tended to equations of higher dimensions, and theref the rule will be found to be true in all kinds of equati< whatever. PROBLEM L To increaae or diminish the roots of an equation by i given quantity** RULE. 1. Take some new letter, and connect it with given quantity by the signs — or +, according as ii required to be increased or diminished. 2 Substitute the powers of tiiis quantity in the eq tion, instead of the powers of the unknown letter, i there will arise a new equation, whose roots will be a mented or diminished as required. * When a cubic equation has two equal roots, it may alway reduced to a lower diaiension, and the solution, by that means, n: mare e»sy. RESOLUTION OF EQUATIONS. 119 EXAMPLES. ^ !• Let the quadratic equation x^ + 8jc + 15 = be given; it is required 10 increase its roots by 7. \ Suftfiose j:= y — 7, Then x^^y^ — 14^4.49 8x= 4- %y — 56 15» 4- 15 y%^^Sy 4- 8s=o=srAff equation required*. 2. Let x^ — /ix2 4- qx — r=io^ be the equation given ; it IB required to diminish the roots by the quantity e, Sufi/iose xsssy +€; Then x^ssyi+Sey^+Sehj+e^ "^ -^.-Z^Ss .^ fiy2 — 2/iey — /le^ I = Oj t^e new equation '\-qxss ^ qy^qc f required^, '^ r sss — r J 3. Let x^+x^ — \0x 4- 8 srsO be given, and let its roots be increased by 4. * For, in the fonner equation jc34-8^+15s3=o, the roots are — »3, and —5, and in the equation y^^-fy-^-SssOt the roots are 2 and 4; therefore the difference is 7» as was required. t The last term of this transformed equation is the same as the ^ven eauation, having e in the place of ^. And ^i<*^ this it appears that, if the last term of any equation is to be destroyed , the difficulty will be no less than that of solving the original equation itself. 150 RESOLUTION OF EQUATIONS. Su/i/iose jc=sy~^4i; Then x3=y3 — l2z/2 + 48i/ — 64 + .r2= 4- y2 — 81/+15 — I0jt7s= — 10I/+40 + 8 = +8 iMdM^Midh*iMi^>^M*^*i 5k« aat/5 — llt/2+30y=«sO, Or 2/^—1 ly+30=30, /A^ equation tequin In ivhich equation y is found aes 6 j anrf conaequ X s=:2. PROBLEM n. To take away the set and tetmfrdm any equation RULE. L Divide the co-efficient of the second term bj index of the highest power of the unknown quantity; 3. Annex the quotient, with its sign changec . some new letter, and this, being substituted for its e in the given equation, will destroy the second tetn required. • * In thiB example, the -given equation is reduced to a quadi sftidin the prrB(.nt case, as well as in ail others, where the last vanishes^ the number assumed (•^) is one of the reots of the pos«d equation The- affirmative roots of an iiquatbn are thangie^ ' ' *-f<i: ones of the sami- \'alue, and the negative foots iiito lf& r . .ve< by only changing the signs of the terms altetnately, bbgumiiig the second. RESOLUTION OF EQUATIONS. 121 « EXAMPLES. 1. Let the quadratic equation x^ — 8x + 15 = be giTen; it is required to take away its second term. Skftfiote x=sy +4(y4-|); Then x»=ya+ 8y + 1 6 — 8x= — 8y — 32 + ISas +15 y* — \=^o=:equatton required*. 2. Let the equation a:^— 9ji:2+26x — 34s=0 be given; it is required to exterminate its second term* %Me X =sy +3(y+|); Then x3=y3^9y2_|.27y+2r ^ 9x«== — 9y2 — 54t/ — 81 + 26^: = +26^4-78 —34 » —34 y3_y«. XQrszo^si equation required. Thus the roots of the equation x4— x3— 19x3-j-49x— 30==o are -j-l, -(-2, +3, — 5 ; but, by changing only the second and fourth terms, the equation becomes x4-f.x3 — 19x3^49x — 30=0, and the toots are —1, —2, —3, +5. All the roots of an equation may also be made afBrmative or ne. gative, by increasing or diminishing each of them by some known quantity. * From this example it appears, that any quadratic equation may be solved without completing the square, by only taking away the second term ; for since ^*=1, or jrsssy/l^l, we shall have jcmey-f- 4^14-4«b5, the root required. And the same may be shown of any otlier adfected quadratic equation whatever. L 122 RESOLUTION OF EQUATIONS. 3. Let x*+8x3 — 5j?2+10jc — 4=0 be given; to ex- terminate the second term*. Suppose X =y — 2(y— |); Then ar4=z/4_8i/3+24i/2— 32y+16 + 8x3= -|.8t/3— 48i/*+96y — 64 — 5^2= — 5y2+20i/ — 20 4-10x= 4-101/— 20 — 4 = — 4 ^ 1/4 — ^292/2+941/ — 92 =^o = equation re* quired, 4. Let x^ — /2x3+yx2 — rx4-«=0 be given ; to exter- minate the second term. P Suppose x^^y'\'-^\ 4 Then ^=y*+fiy^+ _LL + £| + ±- rp ^ ^ + &• = 4- « • Since the sura of all the roots, in any equation, is equal to the coefficient of the seccnd term, it follows that, when the second term is wanting, the equation has both affirmative and negative roots, and that the sum of the affirmative roots is equal to the sum of the negative ones. Thus, in the cubic equation x^ — 7xssi6, .the three roots are -f-3, — .J2/ and — 1, where it is evident that 3ss2-|-l. RESOLUTION OF EQUATIONS. 123 PROBLEM HI. Tojind whether some or all the roots of an equation be rational; and, ij'aoj what they are, RULE*. I. Find all the divisors of the last iermy and substitute them one by one for the unknown quantity. 9. Then, if the positive and negative terms destroy each other, the divisor, so substituted, will be one of the roots of the equation. 3. But if none of the divisors succeed, the roots are, for the general purt, either irrational or impossible* JVbte. When the divisors of the last term are too nu- merous, they may be diminished by changing the equa> lion into another, whose roots are augmented or decreased by a unit, or some other known quantity. EXAMPLES. 1. Let ^3.^4 jp2 — 7j7 + 10 = be the equation pro- posed . Then the divisors o/*(10) the ia6t term will A^ + 1, — 1, + 2, — 2, +5, — 5, + 10, — 10. * Since the last term, in any equation, is always equal to the pro- duct of all the roots in that equation, those roots must, therefore, necessarily be found in the number of its divisors. But this, it is evident, can hold only when the roots are commen- surate, or whole numbers. 134 RESOLUTION OF EQUATIONS. / And theae^ being aubatituted aucceaaively inateat will give I-^ 4— 7+10= — .!_ 4+7+10= 12 8 — 16 — 14+10= — 12 _8_ 16+14+10= 125—100 — 35+10= There/ore + 1, — .*, and +5 are the three roota equation required. 2. Let y^ — 4y3 — 8y + 32«0 be the equatio: posed. 1« Change it into another^ the number of whoae d, ahall be Uaa; thua^ Su/ifioae y=x + 1 Theny*=^x^+4x^+ 6x^+ 4x+ I — 4y3= — ix^ — 12x« — \2x — 4 — By = — Sx — 8 +32 = +32 ac;^n,.^^x^'^l6x+2\ssiO=snew equati 2. The diviaora of the laat term (21) o/* this neiv tion are 1,— 1,+3,— 3, +7,— 7, +21, —21. And if these be subalitnted successively itisttad of a/iall have ■ 1 — 6 — 16 + 21= 1 — 6+16 + 21=32 81 — 54—48 + 21= 81—54+48 + 21=96 tS^c, where none of the others sr, So that 1 and 3 aie the only rational rootay the oih, being im/ioaaible. * Note, The divisor of the last term of this new equation diiQinished in the same ffiscnner as before. RESOLUTION OF EQUATIONS. 125 3. Let x^'^-Sax^''^a^X''-^\2d^s=siOf be the equation Bi*oposed. ffere the numeral divisors of the last -term ( 1 2a^) are I, _i, +2, —2, +3, ~3, +4, -^4, +0, —6, + 13, — 12. -^^nd by substituting these successively instead ofx^ we sMll have 1+ 3— 4 — 12=— 12 —1+ 3+ 4—12=— 6 8+12— 8 — 12= —8+12+ 8 — 12= 27+27-12-12= , 30 —27+27+12—12= Therefore the three roots are 2o, — 2flr, and — 3a. PROBLEM IV. To discover the roots ofequatioyis by Sir Iaaac Newton'* method of divisors. RULE. 1. Instead of the unknown quantity, sul>8titutc sue- ^ssfively three, or more, terms of the arithmetical pro- ^t*ession 2, 1, 0, — I, — 2. 2. Collect all the terms of the equation into one sunr, ^\i«l place them, together with their divisors, in per- feendicuhir lines, ri^ht against the corresponding lerms^ ^i* the pi'ogression 2, 1, 0, — 1, — 2, 3. Seek amongst the divisors for an arithmetical pro- gression, whose terms correspond with the order of the terms 2, 1, 0, — 1, — 2, and whose common difference IS either a unit, or some divisor of the co-cfificieut of the . L 2 i 126 RESOLUTION OF EQUATIONS. fiighest power of the unknown quantity in the given equation. 4V Divide that term of the progression, thus found, which stands against the term in the first progression) by the ratio or common difference* • » 5, To the quotient last found, prefix the sign + or — r according as the progression is increasing or de- creasing, and this number being substituted for the un- known quantity, will be found to be one of the roots of the equation. JVbte. When there is more than one progression^ the roots must be taken out of each. 1 EXAMPLES. U Let x3 — x^ — 10x+6=5s0, be the equation pro- posed. Thetiy by aubatituting aucceaaively the terma ofthefiro* greaaion 3, 1,0, — 1, inatead ofxy the work will atand aa Jbllowa : \at firog, 2 1 — 1 reaulta. —10 — 4 + 6 + 14 diviaora, 1.2.5. 10 1.2.4 1.2.3. ^ 1.2.7. 14 2rf. t^rog 5 4 3 2 jlnd — 3, the term atanding againat o^ being aubaiitutedfor Xy givea^27 — 9 + 30+6=0; and therefore --^3 ia a r^ot of the equation. 2. Let 2;r3-.— 5:c2+4j; — 10=0, be the equation pro- posed. Theuj by aubatituting aucceaaively the terma of the fir o^ greaaion^ 2, 1, 0, —1, — 2^inatead qfx^ the work will atand aafoUowa: RESOLUTION OF EQUATIONS. 127 \8t fir og., results. — 6 — 9 — 10 —21 — 54 divisors. 1.2.3. 6 1.3.9. 1 .2.5. 10 1.3.7.21 1 .2.3. 6.9 2d firog 1 3 5 7 9 Utftro. results. 2 70 1 144 180 •-r 160 —2 90 progressions. Here 5f the term standing against o, being divided by 2, the common difference^ gives 2^\ and this being substitut' fd for xj gives S\^ — 31^+10 — 10= o; and^ therefore^ ^\iMa root of the equation, 3. Ut j:^+x^ — 29x* — 9x-f 180r=o be the equation proposed*. Tkeuy by substituting as before^ the work will stand as foUovn: divisors. '1.2.5.7.10,14, isfc, 1.2.3.4. 6. 8, ^c. 1.2.3.4. 5. 6, ^c. 1.2.4.5. 8.10, isfc. 1.2.3.5. 6. 9y ISTc. ^^ that here are four progressions^ and the numbers 3, 4, ■*-3, and —5, being each substituted for x, make the whole ^9^tQtion vanishy and are therefore the roots required. 4. Given a.^ — Sa-^ — 46x — 72=0, to find the values of ^'» by the method of divisors. j^jis +9^ — 2, and — 4. 5. Given x* — 4>x^ — 19:c2+46x+ 120=0, to find the Values of Xy by the method of divisors. j^ns. +5, +4,-3, —2. 1 2 2 3 5 4 7 6 3 4 3 5 4 5 2 4' 3 5 6 I * Several other rules for discovering the roots of equations may ^ found in Newton's amd Madaurin's Algebra. 1Q» RESOLUTION OF EQUATIONS^ PROBLEM V. To find the foots of cubic equations^ according to the me* thud q/*CARDAN. RULE*. \. Take away the second term of the equation, By- problem second, and it will be reduced to this form: 2. Substitute the values of a and by with their proper signs, in. the following expression, and it will give the root required. Thus : r^uired. * The rule, from whence this method is derived, is xss ^i^+%/(i^+-5V«3) + ^i^^s/iii^-^-^y. and the investigju tibn of it is as follows : Let the equation, whose root is required, be x^^ax^^b. And assume J' >|-2=r:x, and Sj'Zs — a. Then, by substituting these values in the given equation, we'shall Ivave;^3_|-3;a24-3;'«2+«34.aX(;'+2)=;'34-23-f37«X(7+2)+fl And if, from the square of this last equation, there be taken 4 times the cube of the equation yz^:^ — ^a, we shall have^fl 2^3^^ But the sum of this equation and {y^^z^ss^b) is 2y3--.4 i • (52-|-^yi3) and their difference isSz^sa^— ^(^«|. 4a3)j wlience y is found =g^^/>4.^( ;^i>a-f ^i^,and g=a RESOLUTION OF EQUATIONS. 129 ^ote*. When a Is negative, and -^a^ is greater than ^a, the solution, by this rule, cannot be generally ^*tained.'. - / EXAMPLES. . i. Let y3+3y«+9y=13, be the equation proposed; It is required to find the value of y* !• In order to destroy the second term^ letyssx-'^l ; then y»aBBx3 — 3x2+3jr — 1 3ya= +3j:«— 6:c+3 9y = +9ar— 9 x^+6x— 7=13 or, j:3+6xa=20 And from hence it appears, that y-^-z* or i ts equal x, is = ^iA+^(i*»+ 1 a3) + ^i^— ^(i^-f 1 a3), which is the theo- Or, since « is ss — A it will be j>+a;s==)f— -2L«_v 3;^ ^ 3y — '^^A-j-^(^^4- 1 tfS)— — "^^ , f Ae tffltme as the This method of solving cubic equations is usually ascribed to Oardan : but the invention is not his. The authors of it were Scipio ^trreiu, and Nicholas Tttrtalea^ who discovered it about the same ^ime, independently of each other, as is proved by M. de Montucloi Ui his JBUtoire des MathSmatiques. • This is called the irreducible case ; and, notwithstanding several of th^ most eminent mathematicians in Europe have attempted the Bolotion of it, no general rule has yet been discovered. The usual method is by a taWe of sines, or by throwing the ex- pression into an infinite series, and finding the sum of a certain number of termst according to the degree of exactness required. ■\ 130 RESOLUTION OF EQUATIONS. S* jFor a fiut 6, and for b %0, and wis shajii have ^=^|6+^Q.2-f^V«^)--- 4« 2 -r\^v -1- J ^10-1-^(1004-8) ^104.10.3923-^|7=====3 «^20.3923_ 2 2 =2.732— ——-=2.732— .732 =2; /Aa/ m, ^20.3923 2-^32 ji:=:2, and consequently y=sl s=root requited, 3. Given j:3 — 6x= — 9, to find the value of j:. Here a=s — 6, and b = — 9 ; and therefore we shall have i" ^=n*+v?a^^+^a^'-:^^^^p^3) — 2 — 2 — -=-— 1— .2=— 3 ; fAaf w, ar=— 3= roo^ required. EXAMPLES FOR PRACTICE. 1. Given a:^—- 6x^4- 10x=s8, to find x. Ans, x=4. 2. Given y34-30y---n7, to determine y. An%^ ys=3. 3. Given . 1/3— i.36y=91, to determine y. ./^n«. y=7. 4. Given y^ — 3j/=18, to determine y. vf;?*. y=3. 5. Given y34-24y=250, required y. ./^w*. y=5.05. d. Given y« — 3y4 — 2y*— 8=0, to find y. Ann. y=2. RESOLUTION OF EQUATIONS. 13 1 PROBLEM VL Tojind the roota of biquadratic equationay according to the method o/Des Cartes. RULE*. 1. Take away the second term of the equation by problem 2, and it will be reduced to the form x^-^-qx^-^ 2. From the cubic equation y^+'2qy^+{q^^-^8)y — r^ =0 take the second term, and find the value of y by the last problem. 3. Let e be assumed =x/y,/=j7+^e2 — and ^= 2(? *^+4^*+^' 4. Find the roots of the two quadratic equations x^-^ <rj-4/=0, and x^ — ex-f-.§-e=0, and they will be the four roots of the biquadratic required. • Irvceftigation of the rule. Let the given equation x^-|-9'^+^ -f^=sO be equal to the product of the two quadratic equations x^-f- Then, by equating the homologous terms, we shall havey-f ^ — r t^ssqt eg-^fssr, zndfg^^s; and therefore /=? 9+ ^ea—--, g=: k+i««+^. and s:=:fxg=:qi+iqe2+le4^^. Aq4 from this last equation we shall have e^^^qe*J^(jg2 — is)x d^^sT^-ss to a cubic equation, in which the value of e may be fdtfnd, as in the last problem. 132 RESOLUTION OF EQUATIONS. EXAMPLES. 1. Let z^— 4z3 — 8z+33sbO be the equation pr in which it is required to find the value of z. 1. To take away the second term^ let x+ l=z; 2:4=J74 + 4j73+ 6J7»+ A,X+ 1 — 423 — — 4 j;S — 1 2x« — 1 2x — 4 ~.8z 8 — 8x — 8 • +32 SB +32 j?4.*_ 6j?2 — 16X+21 =0, or T/3 — 1 2ya — 48y — 256=xO,/or *Ae cm^/c e< 2. To take away the second term from this egtui fi+4i=sy; then y3^fi^+i2/i^+4Sfi+ 64 — 12z/«= _12/i»— 26/i— 192 — 41y = —4>Sfi—\92 — 256 = — 256 /i3— .96/^=576 Bi|t/(=ig+ic2-^) and g (=ri7+ie»+-^) arc als( and therefore the roots of the quadratic equations x2+ex and x2— «x+^=so, may be determined, and are the four the biquadratic equation required. ^ E. L Note. The co.eificient of x is put equal to e, in both t tions, because, when the second term is wanting, the su positive roots is always equal to the sum of the negative oi a contrary sign. THis rule has sometimes been ascribed to Dm Cartes^ a: times to Bomdelli, an Italian ; but the original inventor « Xouis Ferrari. RESOLUTION OF EQUATIONS. 133 S. Tojind the value offi^ by Cardan's ruiefor cubic ^mtiona. —32 A 1 g ^^^fii and therefore ^=16, or^yssi4y /= -— -\ . 16 , — (5 16 16 f^7jandg=sS. ^ Inthe method oiDes Cartes, above explained, all biquadratic equa- tions are supxxwed to be generated from the multiplication of two Jtiadratic ones : but, accoiding to the following way, which is taken ntmi ^mptof^s Algebra^ every such equation is conceived to arise by taking the difference of two complete squares. Here, the general equation x^-f.a)^4'^^~l~^^H~^=^ being pro- pOied, we are to assume (x8-f-ifl3f+ a)^ — (Bjc+c)2=rf(r*-^flx3^c* '\4 ; in which a, b, and c, represent imknown quantities to be detemuiied. Then x*-f ^a*+A, and bx+c being actually involved, will give x^+oxa+SAx* "1 -f ia»x24-aAx4- A* S. =sx*-j-ax3+^xa — B«jc2— 2bcx— c« 3 -^cx^d: from whence, by equating the homologous tenns, we shall aave j 1. 2A4.|aS -.bS=:6, or 2A-f ja^— A=rB*; 2. aA— 2bc . =sc, or aA— c s=:2bc; 3. aS— cs ssflffOr aS^^/ =c3. Let now the first and last of these equations be multiplied together, and the prodnct win, evidently, be equal to \ of the square of the second; that is, 2A3lf (^a*— A)XAa— ^/a— (ifl«— ^)X</=(b2c2) Bi X^A*-— 2acA-|> cs. M 134 RESOLUTION OF EQUATIONS. 4, To Jind the roots of the two quadratic egua x^^ex-^-fsssOj and x*^— -ex+^=o. :r^+ex+/s=x^+^x+7=:0, x» — ex+g^x^ — 4x-|-3=0. In the Jirat of these x= — ^+\/ — 3, or — 2 — v And in the second J7=i3, and 1. . Therefore^ 3, 1, — 2 + ^ — 3, or — -2 — -v/ — 3, ai four roots of the equation x^ — 6x2— -l6:c+21=o. Jind if unity Be added to each of them^ we shall 4, 2, — \+y/ — 3, and — 1 — y/ — 3, for the roo 24— 4z3— 8z+32=o, the equation fir ofiosed ; the tw of which are im/iosaible, 2. Given x^+2x^—7x^ — 8x4-12=0, to find values of x, Ans. x= 1, 2, — 3, and Whence, by denoting the given quantities ^ac — d, and^c^ (|a»— ^) by i and /, respectively, there will arise this cubic equ A* — ^AxS-j-iA— ^/ssso ; by means of which the value of a nr determined ; and therefore, from the preceding equations, I and c will also be known ; b being found from thence =v^( flA — c Ja2-^), and Css-ge" ' The several values of a, b, and c, being thus found, tha will also be readily obtained : for (x2-|-^i2x-|-a)2 — (bx-j-cJ2 universally, in all circumstances of x, equal to x^-f-ox^-f-ox -^df it is evident, that, when the value of x is taken such th latter of these expressions becomes equal to nothing, the formei likewise be equal to nothing ; and consequently (x2-j-Jax-j-j (Bx-|-C)2. And therefore, by extracting the square root of both sides < equation, we shall have x»4-ia^-f-A=d:Bx±c ; or x=±i a±l (ifl±^B)2dtc— a],*=:±4b— ^a± (TV«^±iaB-f-iB2±c- which exhibits all the different roots of the given equation, ac ing to the variation of the sigp:is. RESOLUTION OF EQUATIONS. 135 3. Given x* — 25x2+60x — 36=0, to find the values ^^^' Ans. 1, 2, 3, and — 6-. 4. Given y^ — Sy^+XAiy^+Ay — 8=0, to find the values ofy. jlna. y=3 + ^5, 3 — -v/5, 1+^/5, awe? 1 — ^/3. 5. Given x^+ \2x — 17=0, to find the values of x, -f«». j:=ix/2±v^( — 3v^2 — J), and also — i\/2± ^(3^2— j). 6. Given x^-^^^x^-^Sx'^ — 4a:+ 1=0, to find the va- lues of x. ^ — 1±^/— 3 ^5±x/2l ^ns, x= < and • , 2 ' 2 7. Given x* — \Ox^+SSx^ — 50a:+24=0, to find the Values of X, Ans, l, 2,3, and 4. This method will be found to have many advantages over that jiven above. In the first place, there is no necessity for being at the trouble of exterminating the secoM term of the equation, in or- dcr to prepare it for a solution ; secondly, the equation a3 — ^^a^-j- ^A— ^/=:o, here brought out, is of a more simple form than that derived from the former method : and, thirdly, the value of a in this equation will always be commensurate and rational t not only when all the roots of the given equation are commensurate, but also when they are irrational^ and even impossible. Example. Let there be given x^-\-\2x — 17=0, to find the value ofx. Here, by comparing this with the general equation x^^ax^^bx -\-cx^d^o, we shall have a=(7, b=iO, c=sl2, and «^=ail7; and therefore kzs^iae — d^Vr, /=4c8-f-rfx(4«*— ^)=36, and a3— iAA2-fifA— i/=A34.17A— -18=0. And, from this equation, a will be found equal to 1; and there- fore B=(2A4.^a2-^)^=^2, c=^^=^^ =— 3v'2» and x=s±}^2±(iT3v/2— l)*=±is/2:f(±3v'2— i)^. 136 RESOLUTION OF EQUATIONS. 8. Given x«—^j73—58j?2 — lUx— .llasO, to find the values pf X. 9. Given j?^— 3x*— 4:c— -3=0, to find the value of oc, . l±x/3 _l±v^_3 Ana* x^ss. ^ana . PROBLEM VII. To find the roots of equations in general^ by the method of afifiroximation and converging aeries, RULfe*. 1. Find, by trial, a nomber nearly equal to the root required. In some particular cases of this rule, the roots may be found by means of quadratics only. Several other methods of solving biquadratic equations have been invented by different authors ; one of the most ingenious of which is that given by M. Euler, in p. 664 of his EUmena d*Algebre. Equations of five or more dimensions may sometimes be reduced to those of an inferior degree ; but the process will be exceedingly tedious, as no general rule can be given for resolving them. • The rules hitherto given for finding the roots of equations are either very troublesome and laborious, or else confined to particular cases; but this method, by converging series, is universal, extending to all kinds of equations whatever; and, though not accurately true, gives the value sought to any assigned degree of exactness. The method of obtaining the roots of equations by approximation was first made use of by Vieta. RESOLUTION OF EQUATIONS. tST 2. Call the number, thus found, r, and let z be put equal to the difference between r and the true root x, 8. Instead of x, in the given equation, substitute its equal rdizy and there will arise a new equation, affected only with z and known quantities. 4. Reject all those quantities in vhich there are two or more dimensions of z, and the value of z will be found by means of a simple equation. 5. Add the value of z, thus found, to the value of r, and it will give the root required neaHy. 6. If this root is not sufficiently near the truth, re- peat the operation, by substituting it instead of r, in the equation exhibiting the value of z, and it will give » second correction for the root required. EXAMPLES* 1. Given x2 — Sx — 3J=:0, to find the value of j: bjr approximation. The root J found by trialy «> nearly equal to 8"; X<?r, therefore^ 8=r, and r'\-z^=^x ; then x2=r2+2rz+22 _5 j7 =—5 r — Sz — 3l=— 31 r2+2rz — Sr — 5z — 31=0/ 3t4-5r— r» 31+40 — 64 7 ^ , «jr zs= — = — • = — =»6, and x^=^ 2r—5 1 6—5 11 8.6 nearly. Andy again, ifS.6 be substituted in the filacg o/r in the last eqttation, tve shall have 51+5yw.r> 31+43—73.96 ^ .04 _ ^^^^^ ^™ 2r — 5 "~ 17.2 — 5 12.2 * ^ and j:sb8.6+.0033s8.6032 nearly. M 2 138 RESOLUTION OF EQUATIONS. Andy tf this value be again substituted /or r, it will give ZS3.0000077808, and x=b8.603277808 ; and so on to any degree of. exactness* 2. Given x^-\-x^-\'X^^Oy to find the value of x by approximation. The rooty found by trialy is nearly equal to 4; Lety thereforey 4ssr, a«dr+a:=x, theny x»a=r3+3r«z+3r2a+23 X sar -i-z r3+3r2z-fr2+2rz+r+2=a90 ; 96 — r3 — rg — r 90 — 64—16 — 4_ 6 __ ®''' ^'^ 3ra+2r+l 48+8+1 57 ^^' ancf :i7ssb4.1 nearly. Jndagainy ifA,A be substituted in the place ofry in the last equationy we shall have 90— r3-^r« — r 90 — 68.921 — 16.81—4.1 ' sss SS.00283, 3r»+2r+l 50.43+8-2+1 ' and :cssa4.1+.00283sa:4. 10283 nearly ; and so on to any degree o/ exactness required. 3. Given ^2+20^:= 100, to find the value of x by approximation. Ans, xs=4.1421356» 4. Given x3+10x2+5ar=*2600, to find the value of ^x by approximation. Ans. 1 1 .00673. 5. Given x^+2x^ — ^23j7 — 70=0, to jfind the value of X, Ans» XS5.1349.. 6. Given x^ — \Sx^+^x — 50=0, to find the value ofx. ^ ^n«. xsss 1.028039. 7. GiVen x^ — Sx^— 75x=10000, to find the value of JC. jSns, xas7lO;2615. a. Given x»+2x4+3j?«+4*»+5x«a54321, to find the value of x. Ana. j3aPB8.4i44. > RESOLUTION OF EQUATIONS. 1S9 RULE II. 1. Assume the general equation az+bz^+cz^+dz^^ &c, =ssfi; where z is the converging quantity, and a, b, c, dy &c. co-efficients whose values are known. 2. Then will ^ be an approximation of the secoftd degree. b c 3. And, if 9 be put == —, we shall have a (a + afi) X A r . r • , . , • for an approximation of the third de- gree. 4. And, in like manner, if w be put = h 7- , a d* -r ac then will — ^ ^ . ^^ . — V ' V / 7— r« ^ an ap- aX(a2+(6+ow)x/i)+(w — «)x/i2 *^ preximation of the fourth degree, &c. EXA.MPLES. 1. Given a:«+20x=slOO, to find the value of a:. The rooty found by trials is nearly equal to 4; Lety therrforey A+z^sXy andy byaubatitutionytheequa* tion will become 28z+2*=4. Whencey from the ruley a ss 28, d = 1, c::sOy i^c. and aft 112 28 Therefore z«= --i-— .»---=»—-««:. 142 1 3 for the ^ a^+b/i 788 197 ' frst approximation. }40 RESOl-UTION OF EQUATIONS, * b c 1 j^ndj since 9 = — — . -— s= — , ^e shall have z := . fl 6 28' {a+sti)y.fi ^ (2 84-|)X4 ^ 28+ n^ ^ «^+(^+tt«)X^ 28X38 + (I + 1)X4 28*-f-4-|-2 -- — = .14213564yor ^^e second approximation, 1386 2d flrf — he 1 wf/zcf. m /z^tf manner % since «;= 1 = — , it ' ' a^ b^ — ac 14 afiy.{a+wfi) will be z =s ' — 7-^-7-,, , s .V . . s — « — aX(a«+(<&+aw)X/i)+(w — *)X/i2 28X4X(28+f) _ 28x(28+f) _ 28X198 _ 28x(r84+12)+^ "" 7x796+4 ""49X796+1 "" 5544 = .1421356236; or or = 4.1421356236,- /or me 39005 ' •^ roo^ required^ extremely near, 2. Given x2+4.r — 10=100, to find the value ot x. Ans. 8,677078. 3. Given x^— 17a:*+54jc=S50, to find the value of x, jina, 14.95407. 4. Given x^ — 2x — 5=0, to find the value of x. jins. 2.094551. 5. Given 2y4— 16i/3^40i/8— 30y=— 1, to find the value of y. ^n^. 2/=:l,284724. 6. Given x«+2x4+3a:3+4x«+5a:=54321, to find the value of x. Ans, x=8.4 14455. In the same manner the roots of other equations may be approxi- •mated; but, to avoid trouble in preparing the equation for a solution*, all such powers of the converging quantity z^ as would rise higher than the degree or order of the af^rozimaiion you intead ta work by. may be f;^r^ where negleded. ^ RESOLUTION OF EQUATIONS. 141 I PROBLEM VIII. To extract the root of any pure fioioer in numbers^ * RULE*. -• 1. Let mmn the number whose root is required ; r= nearest root which can be found by trial ; and na to the index. 2. Then, by putting v* -, we shall have * a* r -4 1— J i— ^ =root, nearly ; or :p «= r+ rxC2v+n) ^ , :rr7z — rx :^\ ^ \ — r — rr rr extremely near, ^X(2i;+a;i— l)+iX(w— l)x(2n— 1) ^ EXAMPLES. 1. Given x^^2; or, which is the same thing, let the square root of 2 be found. Supfioee the root found by trial to be 1.4 ; then we shall L ^ , . 2X1.96 ^^ have m«s:2. r=l.4, 72=s=2, anc? v=s -— =98. ' ' ' 2 — 1.96 * One of the most convenient ndes for practice, which has yet been discovered, is the following : X.'^•\-l)m-^^(n^l)v : (n-f 1)n-|- (fH- l)rn : : r : the true root nearly. Where it may be observed that n^ given number ; rss nearest root, found by trial ; and n=s index, as before. 142 RESOLUTION OF EQUATIONS. j^nd^ therefore. x=tr4 ; --s=1.4+ lli^i££l=,.4-H-i2L.= ,.4 + -i£^= 1.41421356 = 98X594 ^70X198 ^13860 root required nearly. jlnd if the second afifiroximation be uaed^ the root will be found =zlA\42\356236i which is true to the last filac^ of decimals* 2. Given x^^BOO ; or let it be required to extract the cube root of 500. . Sufifiose the root^ found by trial, to beS; then we shall 3 V 5 1'2 ' harve ^=500, rs=8, n=ssS, and v=s _.ssb— -128; Mdy therefore, x^r+ ^^^ . f^^t "^"^^l =8— .063= ' *^ ' vx(6v + 4>n) — 2 7.93 for the first afifiroximation. Or x=:r4. rx(2T;+w) ^^_ ■*"(2i;+2/i— Oxt'+^XCw— 1)X(2« — 1) 6072 — —-=7.937005259936 ybr the second afifiroximation, which is true to the last place of decimals. 3. Let it be required to find the cube root of 2. Ans, 1.259921. 4. Required the cube root of 117. Ans* 4.89097. 5. What is the sursoUd, or 5th rool, of 125000 ? Ans. 10.456389. 6. It is required to find the 7th root of 100000. Ans. 5.1794746792. 7. It is required to find the S65th root of 1.05. Ans* 1.00013366. RESOLUTION OF EQUATIONS. 143 PROBLEM IX. To find the root of an exfionential equation* RULE*. 1. Find, by trial, two numbers, as near the true rodt as possible, ancf substitute them in the given equation uistead of the unknown quantity, marking the errors i¥hich arise from each of them. 2. Multiply the difference of the two numbers, found by trial, by the least error, and divide the pixxiuct by the difference of the errors, when they are alike, and by their sum when they are unlike. 3. Add the quotient, last found, to the number be- longing to the least error, when that number is too little, and subtract it when too great, and the result will give the^true coot nearly. 4. Take this root and the nearest of the former, and, by proceeding in like manner, a root will be had still nearer than before ; and so on, t^ any degree of exact- ness required. EXAMPLES. 1. Given j;*=100, to find the value of x by approxi- mation. By the nature of logarithms xXlog, x=slog. 100=2. ./#«</, since x is found by trial to be greater than 3, and less than 4, Letj therefore^ 3.5 and 3.6 be the two su/i/iosed values ofx. * The rule for solving exponential equations was invented by '^• yean Bernoulli, and published in the Leipsic Acts, 1697. 144 RESOLUTION OF EQUATIONS. Then the log. xsslog. 3.5 =.5440680; and xx log. xsss 1.9042380 2 — .0957620= r«r error^ too Utile. And the log. x=slog. 3.6=.5563025; a?id X X log, x=s2.0026S90 2 , , 4 .0026890BB2e/ error^ too great, \8t number 3.5 Itt error —095762 2d number 3.6 2d error +.002689 QAxadiff. .098451=911111. 0. IX. 002689 ^^„^« =.00273=correc/io». .098451 2d number 3.60000 correction — .00273 3.59727 =x=roo/ nearly. , Againy sufifipae x=3.597; then we shall have log. x=. 5559404, and xXlog,x=sl,9997\76, which, subtracted from 2, gives .0002824, the third error, too little. 2d number 3.600 * 2d error +.0026890 3flf number 3.597 Zd error — .0002824 .003 = diff. .0029 7 1 4 =««»!, ,003 X. 0002824 ^^^^^ =.000285, the correction, .0029714 ' 3rf number 3.597000 corrfC//ow +0.000285 3.597285 =jr=roo; required nearly* 2. Given a:*= 123456789, to determine the value of .i\ Ans, x=8.640O268. [ 145 ] OF INDETERMINATE OR UNLIMITED PROBLEMS. A problem k said to be indeterminate or unlimited ivhen the equations, expressing the conditions of a ques* tion, are less in number than the unknown quantities to be determined. And though such Idnd of problems are capably of in« numerable answers, yet the results, in whole numbers, are generally limited to some determinate number, and may be obtained as follows. PROBLEM I. Tojlnd the values of x and y, in the equation axssby+ c; where a, A, and c are given numbersy which admit of no common drvi%or. RULE*. 1. Let vjh stand for a whole number, and reduce the equation to x«» =wA. hv-4-c dv'hf 2. Make =-£-!j:, by throwing all whole num- a a tiers out of it, till d and /be each less than a. * This nile is foanded on the following obvious principles : That the sum, difference, or product of any two whole numbers 18 a whole number. And that If a number measures the whole of any number, and a part of it, it wiU also measure the remaining part. N 146 OF UNLIMITED PROBLEMS. ^ dy 4- f 3. Subtract -^ ^, or some multiple of it, from ay 2ay Say — , — , , or any other multiple of y, that comes a a a near the former, and the remainder will be a whole num- ber. 4. Take this remainder, or any multiple of it, from some of the foregoing fractions, or from any whole num- ber, which is nearly equal to it, and the remainder, in this case, will also be a whole number. 5. Proceed in the same manner with this last rtfmain^— der ; and so on till the co-efficient of y becomes equal tOHi 1; or^^^=Wi^.=A a 6. Then will y = afi — g ; where /i may be any whol^ number whatever ; and as the value of y is now known^ that of a: may also be found from the given 'wmi^iiip. ■ -w- EXAMPLES. I. Given I9xa=14y — 11, to find x and y in whoK. « numbers. 14y— U ^ ^ 19v Firaty a?= — =5wA.; and -.-^sfp^. 19 1^ ^. z r ^ 19v 14t/—ll 5i+ll _ Then^ by aubtractioriy — — — «a - ■ ■ sk w^* ^ . 5y + H 201/ + 44 20?^ + 6 . ' ^ 30^4. 6 i(ndj by rejecting the 2, — — — • as vfh. 1 y Therefore — ^ — ^ *" "l9" ^* Andy^\9fi-^^\ vfhere^if fi be taken sx\^ fir the leat^ affirmative value t^ y> we •haU have y tm \3^ and x ■■ 9^ the answer. OF UNLIMITED PROBLEMS. 147 2. GWen 3x=8y— 16, to find the values of x and ij iu *hole numbers. ^ 8y — 16 ^ . 2y — I , 2y — I ' 3^3 3 2u I 4t/ — 2 3u ^nd -^ X 2 =5 -^ = wh. But — h also = w//. ,3 3 3 »*,. ^ 4v — 2 3?/ y — 2 -Therefore ^ = — -— = w/r. e=/j. - *^^ence y =± 3/i + 2 ; awrf Ay taking /i =i \j we shall have y ^* 5} and a: == 8, Me awswer. ^. Given 9j; + 1 Sy s 2000, to find all the possible v^c '^ ^8 of X and y in whole numbers. 2000 — I3y ^^^ . 2 — 4y 2 — 4y Or, ^y rejecting 222 -r-y, — r — s= wA. number. 3..4V 4 — 8y -' 9y ^«rf, i X 2 = • = tvh. But* — 19 fl«o=3wA. '9 9 9 TtA X ^y I 4 — 8.7 ?/ + 4 Therefore,. -^ H ^ = = w/i. = /i. ^P hence y = 9/i— 4 ; ant/, 6y taking fi = 1, we «/2fl// /iavt* y asB 5, a/acf jr = 2l5. jindy by adding 9 continually to the lait -value of y, and -^'Ubtracting \3from that of x^ all the fioasible anaivera nviU ^tand aafcllowa : ^ 5 215.189.163.137.1 11.85.59 33- ** i202.176.l50.l24, 98.72.46.20* i 5.23.41.59.77. 95.113.131 y "^l 13.32.50.68.86.104.122.140'''*^ 4. Given 24:p= 13y+ 16, to find x and y in whole numbers. .dns. or = 5, and y = 8. 148 OF UNLIMITED PROBLEMS. 5. Given 14j: as 5y -f ^9 (o find x and y in whole nur bers. Ana, x =s 3, and y = : 6. Given 27x = 1600 — 16j/, to find x and y in whoj numbers. Ana, ?/ = 19, and j;=4i 7. Given 87ar + 256y==15410, lo find the values of - and 2/y in whole positive numbers. Ana. X = 30, and y = 5< 8. Given 5^: + 7y -f 11 z =s 224, to find all the possibl values of x, y, and z, in whole numbers. Ana. The number ofananuera ia 5 ' 9. To determine whether it be possible to pay lOt in guineas and moidores only. A?ia. The queation ia imfioaaibd 10. How many different ways is it possible to pa IQOl. in guineas and pistoles only ? a guinea being equ to 21s. and a pistole to 17s. Ana. 6 different wa^ U. Forty-one persons, men, women, and childr^ spent among them 40s. of which each man paid 4s. eac woman 3s. and each child 4d. how many were there < each ? Ans. 5 men, 3 nvomeny and 33 children 12. I owe my friend a shillino^, and have nothing abou me but guineas, and he has nothing but louis d'ors; tin question is, how must I acquit myself of the debt? tht louis d'ors being valued at 1 7s. Ana. I must fi'ay him 13 guineas, and he must give me 16 iouia d'ors. 13. It is required to discharge a debt of 3511. with guineas and moidores only, so that there may be the kast number of pieces of each sore; and to find what the whole will amount to, when paid every way it will possibly admit of. Ana. The number of ways is 37, and the whole amount 12957. •14. A vintner has wine at 2s., Is. lOd., and Is. 6d. fier gallon : how much of each sort must he take, so as OF UNLIMITED PROBLEMS. 149 to make a mixture of 30 gallons, to be sold at Is. 8d. fier " gllloii? Ane. 16, 2, 12 ; 17, 4, 9 ; 18, 6, 6 ; or 19, 8> 3, o/* each 9Qrt. .^5. To determine how many ways it is possible to pay lOOOI. without using any other coins than crowns, gui- •D^as, and moidores. Ana, 70734 different ways. PROBLEM II. . ^ojind sue A a whole number x, as being divided by t/ie J^^ numbers a, b^ c, ^c. shall leave the given remain^ • RULE* w f* Subtract ~ each of the remainders from a', and ''^'Ue the difference by a, and there will result ""'!; — y, M ' , , fjfc, = whole numbers. ** . a a .^' Call the value of :r, in the first fraction, /i, and sub- f. ^^Ute this quantity in the place of x, in the second frac- . ^. Find the least value of /?, in the second fraction, by ^^^ last problenii and call it r. 4. Let the value of x be found in terms of r, and ^.^bsiituttt this quantity in the place of jc in the third ^^Qclion. 5. Find the least value of r in the third fraction, and ^ull it J? ; and the value of x in terms of *, being substi- '^Uted for x in the fourth fraction, and so on, will give the ^thofc number required. EXAMPLES. 1 . To find the least whole number, which, being di- N2 150 OF UNLIMITED PROBLEMS. Yided by 17, shall leave a remainder of 7; but, beiog^ vided by 26, the remainder shail be 13. Let X ss number required. X — 7 X— 13 Then , and = tvhole numbers, 17 36 X 7 Andy by fiutting = ^, we shall have x = \7fi+ ^t Which value of x being substituted in the ^d Jracti^^i \7fi — 6 , » 26A . , gtves = ivh. But -—- is also =5 wA. * 20 26 AndK therefore^ — — — = — ^z^oh ^ 'z^ - 26 26 Or ' X 3 = = /?-f ^-L — = wA. number-., 26 2o 26 Jlndj by rejecting /i, we have :» wh, =» r. Hence /izss^Sr — 18, and by taking rssl^we shall b> avc And<i consequently^ jr=»17x8-f7= 143, the number required. 2. To find a number, which being divided by II, 19, and 29} the remainders shall be 3, 5, 10. L^t X = umber required, _, a: — 3 X — 5 . X — 10 , . Then — :— ? — r^— ' a"" — rr — = wAo/e numbers, X— 8 vf«ff, by fiutting = /:, we shall have arss 1 1/} -f 3; Which value of x being substituted in the 2d fractiWy l^/_2 , n// — ^ 22/i — 4 OF UNLIMITED PROBLEMS. 151 =s vfh, Andy by rejecting /?, we shall have ^, 3/i— 4 ^ 18/z— 24 18/r— 5 ' 19 19 19 ^ ^ wA. or by rejecting the 1, = wh, number, ^^ —— Wj likewise* = wA. and. therefore^ — L — 19 ' ' 19 *8*~5 fi + S , , , — ■*: = — — — = wh, number, which let be fiut = r. 19 19 ' -* 2%«i,/i=»19r — 5, flnflf:c = (l9r — 5)Xll + 3 = 209r ^52. •^7Z(/, dy substituting this value of x in the Zd fraction^ 209 r — 62 ^ 6r — 4 ^'e shall have -^ = 7r — 2 H 5^= '^^^^ which, 6r — 4 '2^ neglecting 7r — 2, gives — -- — = wh, number: 6r — 4 SOr — 20 r — 20 ^"^1 — :r:r~ X 5 = :» r -( — = wh. or^ • 29 29 29 . . r — 20 ^y rejecting r, = wh, whidh let be fiut = s. Then r = 29* + 20; and, by /netting « = 0, we shall ^iave r =s= 20. Andf consequently^ jr=:209x20 — 52=4128 ; Therefore i 4128 =s7iumber required, 3. To find the least whole number, which, being di- vided by 19, shall leave a remainder of 7; but, being di- vided by 28, Che remainder shall be 13. Ans. 349. 4. To find a nurnber, which, being divided by 3, 5, 7, and 2, will leave the remainders 2, 4, 6, and 0, respec- tively. Ans, 104. 132 DIOPHANTINE PROBLEMS. 5. To find the least whole number, which, being dl - vided by 16, 17, 18, 19, and 20, shall leave 6, 7, «, 9, KimJ 10 remainders. jlna. 333550 6. To find the least whole number, which, beingu divided by the nine digits respectively, shall leave n«z: remainders. jfna, 252(^. -7. To find the least whole number, which, being di- vided by 2, 3, 5, 7, and 1 1, shall leave 1, 2, 3, 4, and 5 fi^i- remainders. jin», 1623. 8. To find in what year of Christ the cycle of the sun was 8, the cycle of the moon 10, and the cycle of indic- tion 10. Ms, In the year 1567, DIOPHANTINE PROBLEMS. * Diofihantine firoblems are those which relate to the finding of square and cube numbers. Sec. and are such as are generally capable of a great variety of answers. They are so called from their inventor, Diophantusy of Alexandria^ in Egyfit^ who flourished in or about the %hi»'d century, and is the first writer on algebra wc iiictt witli amongst the ancients. These questions are so exceedingly curious and ab- struse, that nothing less than the most refined algebra, , * That Diopha7itus was not the inventor of algebra, as has been generally imagined, is obvious; for his method of applying it is such as could only have been used in a very advanced state of the science. Hp no where speaks x)f the fundamental rules and principles, as an inventor certainly would have done, but treats of^ it as an art already sufficiently known ; and seems to intend, not so much to teach it, as to cultivate and improve it, by solving such questions as, befoxe his time, had been thought-too difficult to be surmounted: DIOPHANTINE PROBLEMS. 153 applied with the utmost skill and judgment, can surmount ^be difficulties which attend them. And, in this way, no '^an has ever extended the limits of the analytic art fur- ^«er than Diofihantua^ or discovei^d greater penetration or judgment in the application of it. When we consider his work with attention, we are at a 'OSS which to admire most, his wonderful sagacity and pecu- liar artifice) in forming such positions as the nature of the pit)blems requii*ed, or the more than ordinary subtility of ^is reasoning upon them. Every particular question puts us upon a new way of thinking, and furnishes a fresh vein of analytical treasure, ^bich cannot but be very instructive to the mind, in con- ducting it through almost all difficulties of this kind) when- ever they occur. The following method of resolving these questions will ^ found of considerable service ; but no general rule can be given, that will suit all cases ; and therefore the Solo- mon must often be left to the sagacity and skill of the ieamer. It is probable, therefore, that algebra was known in the world long before the time of Diophmitus ; but that the works of preced- ing writers have been destroyed by the ravages of time, or the de- predations of ignorant barbarians. His arithmetics, out of which these problems were mostly collect- ed, consisted originally of thirteen books ; but the first six only are now extant The best ecUtion is said to be that published at Pdrit, by Monsieur Backet, in the year 1621 In this work the subject is so skilfully handled, that the modems, notwithstanding their other improvements, have been able to do little more than explain and il- lustrate his method. Those who have succeeded best, in this respect, are Vieta, Sraimc^ hr, Kertey, De Bilfy, Ozatmm, Prealet, Samderton, Fermat, and EaUer, The iMt o£ whom, in pBrticuUff has amplified and il- lustrated the Diophantine algebra in as dear and satisfactory a manner as the subject seems to admit. 154 DIOPHANTINE PROBLEMS. .■■■>■ RULE. 1. For the root of the square or cube required, put or more letters such, that when they are involved, ei the given number, or the highest power of the unkn quantity, may vanish from the equation ; and then, if unknown quantity be but of one dimension, the prob will be solved by reducing the equation. 2. But if the unknown quantity be still a squ or a higher power, some other new letters must b( sumed to denote the root ; with which proceed as bef and so on till the unknown quantity is but of one dir sion ; and from this all the rest will be determined. EXAMPLES. 1. 'To divide a given square number (100) two such parts, that each of them may be a sq number. Let x^ (=s □) Ac one of the fiarts^ and then 100- will be the other /lart, ivhich ia also to be a square nun Mf X — 10 had been made the side of the second squa this question, instead of 2x — 10, the equation would have x2.— 20:^+100=100 — x2 ; in which case, x, the^side of th« squaretNeould have been found aelO, and x— '10, or the side c second square =sO; and for this reason the substitution, x was avoided; but 3x — 10, 4xr«-l0, or any other quantity < same kind, would have succeeded eqiutUy as well as the foi though, in some cases, the results would have been less simplt DIOPHANTINE PROBLEMS. 155 •^fsutne the side of this second square = Sx — 10. '^f^vnil 100 — j:2=(2j7 — 10)2=4ar« — 40:c+100; jindj by reduction^ j:=8, and 2x — 10 = 6, Therefore 64 and 36 are the fiarts required. THE SAME GENERALLY. I'tt a'ss given square number, x* (= D) = one qf ita P^ts^ and fl « j f ^rg ^Ae ofAer, which is also to be a square ^*mier. Msume the side of this second square =irx-^aj ihen will fl« — x^ss^rx — a)2= r^x* — ^arx +a*; ^ndy by reduction, x^t . owd ro? — as >, . — . a 8flr* ar'-f-a ar^ — a ^ : v . . — — r-: — =s —-- — = /o a square number. '^er^^ef — — J2a7irf|l— II-J2 ar^ the parts required; ^htre a and r may be any numbers, taken at fileasure, 3. * To divide a given number ( 1 3) consisting of two ^own square numbers (9 and 4) into two other square 'Umbers. If s and r be any two uneqaal nnmben, of which « is the greater; Den w3l Trs, «3— rs, and i^-^-r* be the perpendicular, base, and ypothemise of a right-angled triangle. And from this canon two square numbers may be found, whoee urn or difieicnce shall be square numbers ; for (3r«)s-)-(«s— rs)3iBx ft^r*)*, and (r»+#a)2— (2r*)s»(#«— r»)«, or (#>+rS)«— V .-rS)SaB(2r«)3 ; and this when t and r are any numbers what. * l%it question is considered by Disphantut as a very important net bdng made the foundation of most of bis other probtemi.—Jn 1 56 DIOPHANTINE PROBLEMS. For the side qfthejirat aguare sought jfiut rx- for the side of the secondy sx — 2 ; r being the greater t her J and s the less. Then wHl{rx — ^Y+{sx — 2)aa«(r2a72 — ^rx +S («2a72 — AiSX -f 4)=(r2+«2) j;2 — (6r + Aia)x +13=1 (r2 -f «2)x2=(6r -f 4*)x. 6r-f And from this equation^ by reductionj xisfoimd:=: — — 6r2-(-4r» 3r2+4r* — 3«2 Whence rx—3=z—P—- 3= ^^ , , = r2-f«2 r2+tf2 of the first square sought. ' r2-4-62 r2+«2 of the second* So that if r be taken = 2, and « = 1, nve shall I 3r2-f4r« — 3«2 17 era — 2r2+2*2 6 1 =5 — , and ■ 3= --for the s r24-«2 5 » r2+*2 5 of the squaresj in numbers^ as was required, Ifa^-^-b^ be put equal to the number to be divided^ ge?ieral solution may be given in exactly the same iham the solution of it given above, the values of r and s may be ta equal to any numbers whatever, provided the proportion of tl numbers be not the same as that of 3(a) to 2(A), or 3+2(a+^ 3—2 (a — b). And the reason of this restriction is, that if r a were so taken, the sides of the squares sought would come cut same as the sides of the knDwn squares which compose the g number, and therefore the ojieration would be useless. « The excellent old Kertey^ after amplifying and illustrating problem in a variety of ways, concludes his chapter thus : «< F further account of this ra>^ speculation, see Jndersoau*, Theorei of Vieta'a mysterious doctrine of angular sections ; and likewise rigmiius, at the latter end of the first tome of his Curws Mathe fieus:* DIOPHANTINE PROBLEMS, isT % \ ^. To find two square numbers, whose difference shaU ^ equal to any given number (</). X'ft d be resolved into any two unequal fae tore a andbj. ^ ^ting the greater and b the leaa. •^lao fiut xfor the aide of the lesa square sought j and ^^bssiside of the greater. Then (x+b)^^^x*ssx^ +Ubx+ 6* — x2:ss2bx+b^x^d And if this be divided by bj we shall have 2x+bss:a. a — b Whence^ x = — 3 — asside cfthe less square sought^ a--^b o 4- 6 and j?+d=s— +d3M— ^— aa5«rfe of the greater. So that by putting <fas60, anda+b^2x30, we shaU 30-^3 304-3 Jiave =14, and — ^=16, in- (i4)2= 196, and 3 « (1 6)*=s356ybr the squares in numbers i and so for any dif- ference or factors whatever, 4. To find two numbers suph, that, if either of them be added to the square of the other, the sum shall be a square number. Let the numbers sought Be x and y. Then x*+y*a q, and i/»+xss q, Andj ifr^-'^x be assumed for the side of the first square jp»+y, we shaU have jc«+y""''^^2rx+jc*, or y«r^— .3rx. Whence 2rx9ar*^^ ©r x^» ^ . Againj ify+9 be assumedfar the mde ffthe second square j WfihmUhaPitei/»+ Z7(y*+*)"(y+*)*«»y*+2ty+f«. O 158 DIOPHANtlNE PROBLEMS. r*i— 2rs2 r^ — y r2 — 2r«2 , 2r294.«2 So that and ■ are the numbers 4r*4-J 4r*-|-l quired; where r and « may be taken at fileasureyfirovidei be greater than 2«2, 5. To find two numbers, whose sum aiid diffeten shall be both square numbers. Let a: and x^ — x be the two numbers sought* Theny since their sum is evidently a square nufUbety (w^ic of the conditions of the question will be answered. There remains^ therefore^ only their difference x* — ^a:^ to be made a square. jitidy if for the side of this square there be put j?-i-r, we ^hallhavex^ — 2rx+r^s=x2'^2xy or 2rx — 2jc=r*. tVhenee* J?=r: and x^ — a:= (- j i— . -— -, * 2r — 2 ^2r — 2^ 2r— 2 r2 r2 v2 r2 5o that anrf f ) ~ . are the numbers 2r — 2 \2r — 2^ 2r — 2 required ; where r may be taken at fileasureyfirovided- it be greater than 1. 6. To find three numbers such, that not only the sum of ail three of them, but also the sum of eTery two) shall be a square number. Let 4!XyX^-^4tc^ and 2j:+ 1 be the three numbers sought* Then (4x)+(x«— 4ir)«ap«, <^3--4x)+(2x+ l)issa:a — 2j:+ 1, and (4j?+jr2— 44^+2^?+ l)=a:3+2x+ 1, are . evidently squares, Andy therefore^ three f^fthe conditigniynehtioned in the question aire accomplished. DIOPfJANTlNE PROBLEJyiS. 15? ^^hence i/ remains only to make the quantity (4j?) + *^+ 1), or 6a: + 1 ass /o a square, "^er, therefore^ 6x4- 1 =a2; and we shall have x = •^«</, consequently^ , j — : — j* — 4a2— 4 /a» — 1\ 4a2— 4 ??*•— 2 ." 2-2—2 a4 — 26/^2 4- 25 024.3 ^- +*' ^^— r-' — ^36 — ' '^"^ --r"*""" '*« numbers required: where a may be taken at /lieaauref ^fovided it be greater tfian 5. J^, To find three square numbers such that the sum ^ ^very two of them shall be a square number*. Let a:*, y*i andz^ be the numbers sought ; 7»fnx8+22=a, T/»+22-= D, and x^'{-y^is:iU. ^ X «2_ 1 y r2_ 1 ^ndy hy fiutting — ^: -: , and — = — - — ^ we shall 7m jis z ^r 2* '^ 4«2 ' 22 ^ 4r2 ^vMch are both evidently squares; and there/ore it remains x^ y* o«/y to make 1- — =s square number. * This question is capable of a great variety of answers ; but the least roots, which have yet been ^und, in whole numbers, are 44, lir, and 340. See El4men» d^Algebre, par M. Euler, tome 11, page 327; which is a work particularly calculated for the use of those who wish to obtain a knowledge of algebra without the assistance of a master. 160 DIOPHANTINE PROBLEMS. ■i -^i^s " \ ^ • ssiaqtiare JVd. X (r+ l)*x{' — 1)®=3 /o a square number* And^ by making r — 1 =«+ i> ^^ raB«4-2> w? «Aa// Aor (« +2)2 X(« + 1)«X(«— 1)*+«2X(»+3)2X(«+ 1)»= fo ■ square number. Or («+2)»X(* — l)»+«*X(«+3)»=- 2*4+8»3+6fi2--i» -f 4 as ^0. a square number* J^ovf^ let the root qf this square be <M*Mmtfrfa=|«*— •-! — Then 2s<+ Ss^+6s^ — 4/»+4«(J*a— «+2)«a=ff«^ f.3^5«*+«» — **-f4; or ^44.843„2j^4 — 1,3. or 2«-H ^^tf«« «r«^ — 34, ancTras — 22. ^>— I 575 ^ y r2_l 483 And X = ■■ =s p > , cncr -^ = ass / 2, 48/ z 2r 44 ^ 575 z ^ 483z 48 ^ ^ 44 /w order ^ thereforey to have the answer in whole nuni' bers, let z == 528, and we shatt have x = 6325, and y = 5796, or 528, 5796, and 6325 Jbr the roots of the squared required. 8. To find a number, x, such) that x + I and x — l shall be both square numbera. Ans, xss^ 9. To find a aumber x such, that jr+ 128 and :r+ 192 shall be both squares. Ans, x^ss97* 10. To find a number x suchi that x^ + x and x^ — x may be both squares. Ans. f| DIOPHANTINE PROBLEMS. 161 11* To find two numbers, a: and y^ such that x + y^ **+yi and y^ + x may be all squares. jiiis, x= J, and i/=^V' 13. To find three square numbers in arithmetical pro- S'Cision. jlns, 1, 25, and 49. 13. To find three square numbers in harmonica! pro- i^rtion. .4n8. 1225, 49, and 25. , U. To find three numbers in arithmetical progres- sion, such that the sum of every two of them may be a ^Uare number. ji?is. 120-J-, 840^, and 1560^-. 15. To find three numbers, such that, if to the square ^^ each of them the sum of the other two be added, the ihree sums sliall be all squares. ^na, J, ^, and I, 16^ To find two numbers in proportion as 8 is to 15, ^i)d such that the sum of their squares shall make a ^uare number. jins, 576 mid 1080. 17. To find four numbers such that, if a square num- »*er (100) be added to the product of .every two of them, ^he sums' shall be all squares. ^na, 12, 32, 88, and 168. r 18. To find two numbers such that their difference may be equal to the difference of their squares, and that Ihe sum of their squares shall be a square number. ^na. ^ and ^. 19. To find three numbers in geometrical proportion, such that every one of themi being increased by a given number (1^)) shall make square numbers. .4n&. 81, -J, flw^ifl^. 20. To find two numbers such that, if their product l)e added to the sum of their squares,- it shall make a square number. ^na. S and 3, 8 and 7, IG and 5, ^c, 21. To divide a given number (10) into four such parts, that the sum of every ilirec of them may be a square numl>er. .^»'f. Ij 6, ^^^ and 5fJ 2 162 DioPHANtlNE PROBLEMS. f2. *To find three square numbersi such that their sum, being severally added to their three sides, shall make .square nuipbers. •^'**- iHjh mt^ «icf f}f|Ji==r©o;* required. 23. To iirid two nunit>ers, such tiiat their sum, being increased and lessenedf either by their difftrencei or the di{rei*ence of their squares, the sums and remainders shall be all squares. jina. ^ and -^ 24i, To find two numbers' such, that not only each number^ but also their sums and their difference, being increased by unityt shall be all square numbers. ^n9, 3024, and 5624 25. To find three numt>ers such, that whether theii sum be added ixh or subtracted fromi the square of eacli particular number, the numbers thence arising^ shall be all squares. Anit, y^*, '^*, and y/. 26. To find three square numbers, such that the sum of their squares shall also be a square number. J[n9. 9, 16, and ^. 27. To find three square numbers, such that the dif- ference of every two of them shall be a square number. Ana. 485809, 34225, and 23409. 28. To divide any ^en cube number (8) into three other cube numbers. Ana, ^^-^^ and \, 29. Two cube numbers (8 and I) being given, to find two other cube numbers, whose difference shall be equal to the sum of the given cubes. Ana. ^^ and ^^. 30. To divide a given number (28) composed of two cube numbers (2T and I) into two other cube numbers. j/n,. tf^JJH ««' mum 'A* root,. 31. To find three cube fiundbers, such that, if from every one of them a given number (l:}'be subtracted, the sum of the remaindets shall be » square. * The answers to many of diese ^lestioiis caxmot be given in whole nuinben. SUMMATION OF SERIES. 163 !1. To find tiiree numbers such that, if they lie seve- lUf added lo ihe cube of their sura, the lliree suras 'hence ai'ising shall be all cubes. ^d. lo lind three numbers m ai'itbuietica] proportion, •icli thai the sum of iheir cubes shall be a cube. Jint. 3, 4, S, or 149, 236, 363, We. "4. To lini] three cube numbers, such that thdi* sum 'i^l be a cube number, ^'is. 3^ 4\anrfi3, or 2P, 19^, IB 3, ere ^i. To tind two numbers such that theii' sum shall be 'I'lal to the sum of their cubes. ^ns. tand&. SUMMATION AND INTERPOLATION INFINITE SERIES. Th* docirine of injinilc ecriet is a subject ivbicli has B^g;cd the attentioti of the greatest mathematicians in dl ages, and is, perhaps, one of Ihe most abstruse and difiicult branches of abstract mathematics. To find the sum of a series, the number of ivhosc lerms is inexhaustible, or infinite, has been considered iiy some as a paradox, or a thing impossible to be doae. But tliis difrictilty will be easily removed, by considering lilt every finite magnitude whatever is divisible in rij^- :iim, or consists of an infinite number of parts, whose L^'gregate, or sum, is cijual to the quantity first pro- posed- A number actually infinite is, indeed, a plain con- Crmltction lo all our ideas; for any number which wc can possibly conceive, or of which we have any notion, most always be determinait: and finite i so that u greater may be ntill assigned) and a greDter alter tliis; 1«4 SUMMATION OF SERIES. and so on, without a possibility of ever coming of the increase or addition. This inexhaustibility, in the nature of nui therefore, all that we can distinctly compreheni infinity; for though we can easily conceive th quantity may become greater and greater wit yet we are not from thence enabled to form ai of the ultimatum^ or last magnitude, which is of further augmentation. We cannot, therefore, apply to an infinite f cx>mmon notion of a sum, or a collection of sei ticular numbers, which are joined and added one after another; for this supposes that th( culars are all known and determined. But series generally observes some regular law, and ally approaches towards a term or limit, we c conceive it lo be a whole, of its own kind, an must have a certain real value, whether that val terminable or not. Thus, in many series, a number is assignable which no number of its terms can ever reach, c be ever equal to it : but yet may approach to it manner as to want less than any given difiereni this we may call the value or sum of the series being a number found by the common method tion, but such a limitation of the value of the seri in all its infinite capacity, that if it were possib all the terms together, one after another, the su be equal to that number. Ag^in, in other series, the value has no limitat this may be expressed by saying, that the sum o riesis infinitely great; or, which is the same th it has no determinate or assignable value, but carried on to such a length, that its sum shall ex given number- whatever* SUMMATION OF SERIES. 165 According to the common rule for- summing up a &ute progression of a geometric decreasing series> ^here r is the ratio, / the greatest term, and a the least, ^ sum is (rl — a) -5- (r^i) : and if we suppose c, the ^ extreme, to be actually decreased to 0, then the sum of the whole series will be r/-s-(r — 1): for it is demon- s^le, that tlie sum of no assignable number of terms ^'f the series can ever be equal to that quotient ; apd yet ^ number less than it will ever be equal to the value of the series. Whatever consequences, therefore, follow from the Opposition of r/-^(r^ 1) being the true and adequate ^ae of jLhe series, taken in ail its infinite capacity, as if ^ the parts were actually determined, and added toge- ^r, they can never be the occasion of any assignable pitor, in any operation or demonstration where it is used i<^ that sense; because, if you say that it exceeds that ^alue, it is demonstrable that this excess must be less than any assignable difference, which is, in effect, no difference at all ; whence the supposed error cannot exist, )Dd consequently rl-i-^r — 1) may be looked upon as repressing the adequate and just value of the series, con- inued to infinity. But we are further satisfied of the reasonableness of bis doctrine by finding, in fact, that a finite quantity ictually converts into an infinite series, as appears in the ase of circulating decimal!^. Thus^ -f, turned into a do limal, is = .6666, &c. = ^ + ^ + .^^ + r^j^y &c. ontinued ad injiuitum, fiui this is plainly a geometric eries, beginning from -^, in the continued ratio of 10 to , and the sum of all its terms, continued to infinity, will vidently be equal to |, or the number from whence it iras originally derived. And the same may be shown of many other series, and if all circulating decimals in general. 166 SUMMATION OF SERIES. PROBLEM I. v^/iy series being given^ to find the several order differences, RULE. 1. Take the first term from the second, the sec from the third, the third fi-om the fourth, £cc. and remainders will form a new series, called the first oi ofi differences, 2. Take the first term of this last series from the cond, the second from the third, the third from the fou &c. and the remainders will form another -new sei called the second order of differences. 3. Proceed, in like manner, for the thirds fourth^ Ji &c. orders of differences ; and so on till they termin or are carried as far as is thought necessary. EXA.MPLES. 1. To find the several orders of differences iu the ries 1^ 4, 9, 16, 25, 36, &c. 1, 4, 9, 16, 25, 36, Sec. 3, 5, 7, 9, 11, &c. \st diff, 2, 2, 2, 2, &c. 2flf diff, 0, 0, 0, &c. 2. To find the several orders of differences in the sei 1,8,27,64, 125, 216, &c. 1, 8, 27, 64, 125, 216, &c. 7, 19, 37, 61, 91, &c. \stdiff. 12, 18, 24, 30, &c.2rfflfe;^. 6, 6, 6, &c. 3c/ diff^ 0, 0, 5cc. •^I^ATION OF SERIES. 1 67 To find the several orders of differences in the s 1, 3, 6, 10, 15, 21,&c. jin€. let 2, 3, 4, 5, is^c, 2d 1, 1, 1, IS^c. To find the B^eiral otders of di£Perences in the series ,^0,50, 105,196, &c. ^na/Ut 5, 12, 30, 45, 91, ^c. 2(/9, 16, 25, 36, ere. 3rf 7, 9, 1 1, ^c. 4th 3, 2, ^c. PROBLEM II. arieaj a, A, c, <f, f, ^c. d^m^* grven^ to find the first ttrm of the nth order of differences, RULE*. ;t ^ stand for the first term of the nth differences. • . ^, .. w — 1 n — 1 n — 2. ben wHl a — nb+nx — -—c — nx ■ X — r-flf + — 1 n — 2 w — 3 „ . , t. , -- — X — ^ — X — — e, &c. to n + 1 termsss^, when an even number. id — fl 4- «3 — nx-— — c + wx— T— X — 5— c/ — n x - X s- X e, &c. to n + 1 terms = ^, when n 3 4 odd tia'mber. Vhen the several orders happen to be very- great, it will be converaeht to take the logarithms of the quantities conGemed, I differences will be smaller; and, when the operation is ^ the quantity answering to the last logarithm may be easily 168 SUMMATION OF SERIES. EXAMPLES*. 1 . Required the first term of the third order of diffe- rences of the series 1| 5, 15, 35| 70, &c. Let a, b^ c, d^ e^ life. = 1,5, 15, 35, 70, ^c. and n « 3. ' Then — a + nb — n X c + » X --r— X — -— rf = 2 2 3 — a + 3d— 3c 4- d = — 1+15 — 45 + 35=4 = the Jirat term required. 2. Required the first term of the fourth order of dif- ferences of the series 1, 8. 27, 64, 125, &c. Let a, d, c, d^ e^ ks^c. = 1, 8, 27, 64, 135, ^c. and na=4. ^. ^ . n-i— 1 n— 1 n — 2. 7%tfna — wd + nx— -— c — nx— r— X — -r—d+nX 72 — 1 n — 2 w — 3 , . ^ — — X— T— X e=a — 4d+ 6c — 4rf+ e=l— -3^ + 162 — 256-1- 125=0; ao that the Jirat term of the fourth order ia 0. 3. Required the first term of the fifth order of diflRe- rences of the se<ies 1^ ^, } -^^, 8cc. 4. Required the first term of the 8ih order of diffe- rences of the series 1, 3, 9, 27, 81, Sec. •^n«..256. * The labour in these kind of questions may be often afairidgedbf ^tting cyphers f6r some of the terms at the beginnings ofui€ if* lies; by which means several of the differences will be equal tff^ and thie answer, on that account, obtained in fewer terms. SUMMATION OF SERIES. 169 PROBLEM III. To find the nth term qfthe aeries Uy by r, d, e^ ^c. RULE. Let cfS rf ", flf"S d^y &c. be the first of the several orders of differences found as in the last problem : . 77—1 . . . 72—1 7Z— 2 , . 72—1 Then wUl a+-r— fl^'+-r— X— r-^" +— T" X fi.^3 n — 3 ,„. . 77—1 71 — ^2 77 — 3 n — 4 , Ice. be asnth term required. EXAMPLES. 1. To find the 12th ter^of the series, 2, 6, 12, 20, 30, Sec. 2, 6, 12, 20t 30, &c. 4^ 6, 8, 10, &c. 2, 2, 2, &c. 0, 0, &c. Here 4 onrf 2 or^ the first terms qf the differences ; Lciy therefiire^ 4a(/>, 2&=flf'S and n:=s\2. Thena+'^d^+^^X^^dii^2+\\d' + 55^"»2+44+lU)aBl56a 12rA /frui, or the anstver required, P 170 SUMMATION OF SERIES. 2. Required the 20th term of the series 1, 3, 6^ 10, J 5, 21, &c. 1, 3, 6, 10, 15, 21, 8cc. 2, 3, 4, 5,' 6, &c. 1, 1, 1, 1, &c. 0, 0, 0, &c. Here 2 and 1 are thejirat tenna of the differences. Let^ therefore^ 2=s:d^i !«=:</'", and n=s%Oy 77-^-1 7?— —I 7^— — 2 Th€na+ d^+ X-5-rf"=l + 19rf' + l71c?"=5l4-38+171=210=20/A rtrm required. 3. Required the 15th term of the series 1, 4, 9, 16, 25, 36, &c. ^ J^ns. 225. 4. Required the 20th term of the series, 1, 8, 27, 64, 125, &c. .4n8. 8000. PROBLEM IV. To find the sum of n terms of the series c, by r , d^ e^ Is^c. auLE. Let rfS rf", cf"S tf»^, &c. be the first of the several orders of differences. Then wiU «fl+nx^^d' + n x ^— X --"^^.x <^" w— 1 w — 2 w — 3 72 1 n 2 « + „X ^ X — X — xd-+»x^x V^ 72—3 72-— 4 , —J— X — j-«'^ &c = to the suqi of n terms of th« Befits, SUMMATION OF SERIES. 171 SXAMP^ES* 1. *To find the sum of n terms of the series 1, 2, 3, 4> 5) 69 &c. 1, 2, 3, 4, 5, 6, &c. 1, 1, 1, I, 1, Sec. 0, 0, 0, 0, See* Here 1 and are thejir%t ierma of the differences: Let J therefore^ a=s I, d^sss 1, and e/ii=0; _ / w — I, . «2 — n w2 + n 7%^ lutllna + nx—T-'dissn'^ — = — - — =s sum qfn terma^ as required, 2. To find the ^Su^-iSf»'«; terms of the series 1*, 2^, S^, 4*, 52, &c. or 1, 4, Q^fl^, 2i5;'8cc. 1, 4, 9, 16, 25, &c. 3, 5, 7, 9, &c. '2j /2,'!;j2,"8cc. .. : .;d, 6, Sec,- •. . ^(?re 3 and 2 ar^ the Jtf^t terms of the differences : Let^ therefore^ as=:l, di=3, awrf rfii=2. Then nvillnd +'« X -i^II-rfi+nx -^^ X — l-c/"=72H- • ' - '2 2 3 ^ , n — 1 . ^^ w.i— 1 71.^2 • 3n3 — n w^ — 3««+27i .^2^23 2:^ 2 «X(«+l)x(2n4.l) ^ . - . , .«s i i—^ i SSI sum ofn terms J as required. * Any term of a given series, or the sum of any number of its terms, may be accuratelx determined, when the differences of any order become sftlasrequal to each other. 4 172 SUMMATION OF SERIES. 3. To find the sum of n terms of the series P, 2^, 3», 4^, 53, 8cc. or I, 8, 27, 64, 125, &c. 1, 8, 2r, 64, 125, Sec. r, 19, 37, 61, &c, 12, 18, 24, &c. 6, 6, &c. 0, Sec. Hence thejirat terms of the differences are 7, 12,awd6. Letf therr/orcy aal, d^osTt (/iial2, and d^^^m^e. 2 2 3 n— -1 w-^2 n — 3^,,, . ^ « — 1 , ,^ + WX -r- X -T— X— 7— rf"i«»+7«X-T- +1^ « 3 4 J n — 1 n — 2 . n — 1 n — 2 72^—3 7n*— S» 4 4 =s 5MM ©/"n terms f as required. 4. To find the sum of n terms of the series 2, 6, U 20^ 30, 8cc. , «X(«+ l)XC«+2; 3 5. To find the sum of n terms of the series 1, 3, 6, IC 15, See. ^ n n+l «+S 1 2 3 6. To find the sum of n terms of the series 1, 4, IC 20, 35, Sec. . n Ti-fl n+2 «+S 7. To find the sura of n terms of the series 1*, 2*, S^ 43, Sec. or 1, 16, 81, 256) &c. . n» w4 «3 n 5 ^ 2^3 5 SUMMATION OF SERIES. 1 73 PROBLEM V. TTie aeriea a, ^, c, «/, Cf i!fc, being given j whose terms ^T€ an unit's distance jfroni teach other ^ tojindany interme- ^atc term by interpolation, RULE. Let X ht the distance of any term y to be interpo- la ted, and d\ d", d"i, &c. the terms of the dif- ferences : Then will a+xe/i+arX^^d^+arX^^^X —dni+xy.^— X -^ X -j-rf-', &c. =y. EXAMPLES. 1. Given the logarithmic sines of 1° o', 1° 1', 1° 2', and 1"* 3', to find the sine of 1° 1' 40". To' . l** I' 1*^2' 1°3' Sines 8.2418553 8.2490332 8.2560943 8.2630424 71779 706 H , 69481 —1168 — 1130 38 Here the first termsofthe differences are7l779 — 1 168, mnd 38. Lett thereforcy ar==l*' 1' 40"— 1° 0'«l'40"=r lf= distance ofyy the term to be inter/iolated ; and d^^sT 1779, <f Us 1168, a7i(i</"i=s38. _ ,f X— 1 Then fw/^yaoa-fardi+^X— t-c^"+J?X— ^-X 2 ^ '^'^i/"i-«a+f/i+4t/"— ^"i=»8.2418553+ P 2 ir4 SUMMATION OF SERIES. .0119631— ..0000694 — .0000002 =8.25 375 33 s tine qf 1* 1' 40", a9 was required. * 2. Given the series ^ ^ ^^ ^ ^-j, &c. to find th« term which stands in the middle between i^» and ^. 3. Given the natural tangents of SS** 54^ SS"* 55', 88** 56', 88'* 57', 88** 58', 88** 59', to find the tangent of 88** 58' 11". Jina. 55.711144. PROBLEM IV. Having given a aetiea qf equidistant terms^ a, 6, r, d, f, l^c, whose Jir at differences are small ^ tojind any intermc' diate term by imterfiolation. RULE*. Find the value of the unknown quantity in the equa* tion which stands against the given number of terms, ia the following table, and it will give the term required. f . a — b^» 2. 0^—3*+ c= 3. a — Zb+ 3c — rf= .4. a— 4d+ 6c^ 4rf+ ^= 5. a — 5.^+ 10c — 10flf+ Se — /=0 6. 0.^66+ 15^—20£r+ 15c — 6/*+^«0 n-.-! n— »1 rr i g 7. a-^d+nx— r-c— -nx— -- X— r—e;^ b'c. «aO. 4 3 3 • The move tenos ive givea the mom accucatd/ the eqvatioa will sppraxhMsitv. SUMMATION OF SERIES. 175 EXAMPLES* I. Given the logarithms of 101, 102, 104) and 105 ; to find the logarithm of 103. Here the number oftermBOte 4. Therefore against 4) in the tablej we have a— 4ft+6^-** id+essO; or c== — ^ — ^ o as 2.0043214 Whence 2 *= 2.0086002 irnence ^ ^^ 2.0170333 <?» 2.0211893 4X(d+</)sl6.1025340 a+e SB 4.0255107 6)12.0770233 2.0128372a:/oj'. of 103, at re- quired, 3. Given the cube roots of 45, 46, 47> 48, and 49 ; to find the cube root of 50. ^na. 3.684033. 3. Given the logarithms of 50, 51, 52, 54, 55, and^56; to find the logarithm of 53. jim. 1.7242758695. PROSCXSCUOUS XXAMPLBS HBLATXVO TO SBBIES. 1. To find the sum (^) of » terms of the series 1, 2, 3^ 4, 5, 6, 8cc. Firtt^ l+2+3+44-5,Urr.....fi8B^. jtndn+ (n— l)+(n— 2)+(n— 3)+Ci>-^), tf'c ire SUMMATION OF SERIES. Therefore («+i)-f (n+l)+(n+l)+(n+l), ^c. And consequently (n+ l)xns=25,' or 5= =«ttOT required, ^ 2* To find the sum (S)o^n tenns of the series 1> 3, 5, 7, 9, 1 1, &c. . JF^rsty l-(-3+5+r+9,&c (2n— 1)=5. ^nrf (2«— 1)+(2« — 3)+(2w— 5)+(272— r)+ (2n— 9), e5*c 1=5. There/ore 2»+2«+2«+2«+2n+, Isfc 2n^2S. Andj consequently^ 2nXnsgs2Sj or Sss — . — issn^ss sum required, "* 3. Required the sum (5) of n terms of the series a+ \a+d)+la+2d)+{a+Sd)+{a+4,d)y &c. Firsts a+{a+d)+(a+2d).+(^a+3d)y ^c a+ Qi—l)Xd^S. And a +(nd — d)+a+(^nd — 2d)+a+(nd — 3d)+a+ Qnd—4.d)j ^c a=:5. _ Therefore 2a+(n(/— rf)4^i||fefoicr--^)+2fl+(m/ — d)-, i^c.2a+(nd—d)t=2S. ^ "^'W Andy consequently^ {2a+nd-^)Xn=s'2S; or 5=(2a-^ wc/;— ^)X —-=»««»» required. OR thus: •- . * > - ■ • Firsty a+{a+d)+(a+2d)+(a+Sd)+(a+4,d), ^c. ^ $(+l + l + l + i + l, ^c.xa > o 1(4-0+1+2^+3+4, ^c.XdS "^ JButn terms ^1+1+1 + 1 + 1, ^c.= «, Andn terms o/o+ 1+2+3+4, erc.«^2i!lZli2. SUMMATION OF SERIES. 177 Therefore S:::xna + "^^'^'^^^^^» (2a + nd—d)x n — , aa before. 4. To find the sum (S) of n terms of the series 1, x^ X^y J?3j ^4j gcc. Firaty 1 +a;+x«+a:3+xS tl^c x^^xbS. And x+x^+x^+x^-^-x^^ is^c x^^sSx, Tkerrfore — l+x^saiSx — S; Or Ssss srsaum required. Andj when x ia a firofier fraction^ the aum of the aeriea^ continued ad infinitum^ may befiund in the aame manner* Thuaj 1 +x+x^+x^+x^j ^c.bsS. Jnd X+X*+X^+X^+X'f IS^C.msiSx. Therefore — il ss^— ^/ or S^Sxbs 1. Whence^Sssi xmaum qfan it\fini$e number qfterma^ 1— ^ar aa required. 5. Required the sum (S) of the circulating decimal •999999, 8cc. conttnued ad infinitum. First, .999999, to-c. =^ + ^ + ^ + ^, l:fc. ^ 10 ^ 100 ^ 1000 ^ 10000 1 1 . 1 . 1 «- ^ 10 100 ^ 1000 ^ 10000 9 1 1 1 10.S Therefore X + r^ + r^^^^^^c.^^. T" sT 9 ' Therefore 5=< 178 SUMMATION OF SERIES. 6. Required the sum (5) of the series fl^+(«+^^i (a+2d)2+(a+3cf)2+(a+4c/)«, &c. continued to n tcrnT^ First ^ a2=flS (a+ d)2=a«+2Xlarf+ \d^ (a+2d)2=a2+2x2ad+ 4flf2 (a+3^)2=a2+2x3acf+ 9rf« Ca+4rf)2=sa»+2X4acf+ 16d» e)^c. ere. 'Sumxifn terms q/*(l + l + l + l,^c. Xa* + . . . rfiW 0/ (0 + 1+2 + 3, ere. X2ac/ + . . . cffV/o of (0+1+4+9, ere. L Xc/2 Butn terms q/* 1 + 1+ 1 + 1, £5*c.==n. i>tV/o 0/0+1+2+3, erc.=!2fc:l), •^«rf (/z«o (/0+ 1+4+9, erc.^'i^ii!!!:!!^^":- ^ * 1X2X3 rr^^enci? ^=nxag+wxry2— l)xarf+''^^"~^^^^^""" ^^ 1X2X3 1X2 X3 «wm required. 7. Required the sum (5) of the series a»+(a+c/)^-f (a+2cf)3+(a+3c?)3+(a+4rf)3, &c. continued to n terms. First f a^ssa^ (a+ </)3=.a3+3Xla2c?+3X lflrf«+ Id^ (a+2d)33=a3+3x2a2rf+3X 4flrf2+ . 8rf3 (a+3rf)3=a3+3x3a2</+3X 9acf2+ 27^3 (a+4rf)3=a3+3X4c2cf+3Xl6a</s+ 64rf3 (a+5d)3«a3+3X5o3rf+3X25fld2+135</3 err. is^c* SUMMATION OF SERIES, 179 SuTnofn$erm9o/{\ + l + l + Uisrc.) + ... ditto ©/• (0+1+2+3, ere.) ^ i<> J xSa^d ejore, o-<; ^ _ ^fg^ ^ (0+1+4+9, erSr.) Sad^ H ditto of (0+1+8+27, i!fc.) I terms o/^l + l + l + l, ^fr. ==» ...0/0+1+2+3, ere, =r^^ig=:i^ . . . 0/0+1+4+9, ere. ^!!X(— i)x(2«-i) 0/0+1+8+27+64+125, eTc. = 1X3X3 n3 — 2n3+«2 2X3 o « . «XC»— I)x3a2(/ . equently the sum S^rstiXarA ^ r- h ^ ^ 1X2 *\) X (2w~l) X 3g rf» (n^— 2n3 +n8) x d^ 1X2X3 2X2 *" ruired** Required the sum (4^) of n terms of the serie$ r+15+31, &c. erma of this series are evidently equal to l^ •(*+2), -4), (1+2+4+8), ^c. or the successive sUms of ^netrical firogression 1, 2, 4, 8, 16, eTc. therefore^ a== 1, ancf r=2, and we shall AflVtf a+ar ar^+ar^i eTc. =1 + 2+4+8+16, e^c. an account of figurate numbers, with the methods of find* sums, &c. I must refer the learner to Simpson'i Algebra, Behere he will find this subject fUly explained. 1 80 SUMMATION OF SERIES. But the 9umoflf2y 3, 4, is'c. terms of tfda aeriea 2. t -„ (r«— 1) X -2_ 3. -;;:3p « (r^-1) X j£j *' -TUT ^ ^^'^ ^ 7=i •^ ' r I i — ntermMo/l + \ + \ + ljCrc^ But 1 + 1 + 1 + 1 + 1+1 + 1, e^c. c=«. Whence 5=(r* — 1)X — «)X L « ^unrequired. — r— 1 r— 1 9. * Required the sum of («) terms of the scric^^ 1.3 7 15 ,31 ^ 7%tf terms of the series are the successive sums qfthegeo - — metrical progression ^+i+i+T+-A'> ^^* a a Lety therrforcy fls=l andrs^^y /^ff« tw^c+— +— + TT+7---, ^5*0. continued at pleasure. 64 138 Many qoestiimi of tUt kind, as wcU as several other thiqgi.ie. IsAtig to series in general, may he fonnd in Dodson^ MaflK natkM |v Repository, whkh is an txcd&ai analytical performance. •I SUMMATION OF SERIES. 181 ut the auma of 1, 2, 3, 4, ^c. terma of (his series are (r — I) xa a (r3— 1) xa 1 a (r3_l ) XI , 1 a (r — 1) xr2 ^ 7-2' r — I (r— .1) x»3 ^ r3^ r— 1 IsTc. ^c. Cntermaqfr +r'^-r'{-rjiS*c, r— I / — « terms o/-+ -;+--+ — cJJ'c. JJm^ r+r+r+r, ^c. s=;ir. 1111 r« 1 1 r r2 r3' (/ — l;x/""-^ a 7*»— . t IVhence S=s • x (nr — ; - )=a?/;;z rtt/u/rcd. r — 1 ^ (r — l)Xr"--* ' 10. To find the sum (5) of the infinite series of the ^ciprocals of the triangular numberS) — | f- 1 , <:c, 13 6 10 Let-- — (-—-4.-— + --, Isfc. ad ivjinitiun = S, 1 3 D 10 Or, — + — + -i +J-, eJ'c ^ s. ' 1.1 t.3^2 3 2.5 Then -L + J- + J- + J-, l:fc = — . •1.2 "^ 2.3 ^3.4 4.5 2, '^'f-4)*(T-i)+(T4)+a-i). 182 SUMMATION OF SERIES. Ory 11111 — I 1 1 1 ^c. 1^2^3^4^5 S 1111 J^=T 5 1 Whence y —-=—-; or 5=2= sum required, 11. * To find the sum of n terms of the seru I 1 1 1 1 1 10 ' 15 1 1 1 I 1 1 12^345' » .r,, 11111 1 Then z -= — + _ + f- _, isTc. tc—. 1 2^345 n And r— • f 1 1 1111 l'72+l 2^34^5 n+ • The figurate numbers, of which the terms of this and sevei other series are the reciprocals, may be exhibited thus : {1st. order") ri, 1, 1, 1, 1, &c. 2d. order 1, 2, 3, 4, 5, &c. 3d. order \ are J 1, 3, 6, 10, 15, &c. 4th. order | 1, 4, 10, 20, 35, &c. 5th. orderj U» .^» 15, 35, 70, &c. It may also be remarked, that different series are ^stinguish by particular names, according to the nature of their terms. Thas, a series, whose tenps continually decrease, is c^lled-a \ verging series. And a series, whose terms neither increase nor diminish, is caU a neutral series. Thus: 1— i+i-.i+|, &c. converging ; 1—2+3—4+5, fi diveigiDg: and 1— 1+1— l+i &c. neutral. SUMxMATION OF SERIES. 183 therefore =- + -+— .-| .e^Tcro — . ^ 1 72+1 2^6^12^20' n 72+ 1 ^ n 1.1.1 1 1 Or = h -^, isTc, to . 72+1 2^6^12^20' w.(/i+l) Tf-, 272 1 . 1 . 1 . 1 c^ 2 iv hence = f- , ^c. to -i . 72+1 1 3 6 ' 10 72.(n + l) a= —7— s=su7n of the eertesy or answer required. 72+ 1 12. Required the sum of the infinite series +- _L + -L. + J_, &c. <«*0«4 0»4«d 4*d*D - Let r= ■T- + "r + "Q"^T"*"T'' ^^' ^^ infinitum. Then z— — = ^'T + ■T"^ — » ^^* *^ trans/ioaition, ^72d 1 = h ' h — 4" — 1 ^c. ^y subtraction 1.2^2.3^3.4^ 4.5' ^ Or 1 — — = \ 1 1 , ^c. 6y transfiosition, 2 2.3^3.4^4.5 5.6' ^ vf72cf — = 1 1 T— 9 ^c* by subtraction. 2 1.4.3^2.9.4^3.16.5' 1 2 , 2 . 2.; , 2 2 1.2.3^2.3.4^3.4.5^4.5.6 mence 1-^2=^ + 5^;^ + ^, Isf cad infinitum. ^^, i. ^ 2= i.; therefore -1^ + ^ + ^, con^ tinned to infinity y is equal to — , nvhich is the sum required. Hi 184 SUMMATION OF SERIES. 1 13. To find the sum of n terms of the series 1.2.3 I 1 I « H 1 , &c. 2.3.4 3.4.5 4.5,6 Let r= 1 1 (-, ^c. to — • — ; — -. 1.2^2.3^.3.4^' w.(w+l) Then z-^ — sa 1 1 , Is^c, lo 2 2.3 3.4 4.5 • n.(n+l) 2 (n4-l).(72+2) 2.3^3.4 4.5^5.6 ^*"'«^'""' T- („+i)I(„+3) " rli + d:* + aXi' {contirtued to n terms) by subtraction. Whence — — — * » ■• — , 4 2.(w + l).(»4-2) 1.23^ 2.3.4 3.4.5* i^c» (^continued to n terms)^ by division, *^ndi conaequentlyy + 1 , continued to ia^.3 2.3*4 3.4(5 n terms, is^^ — — -— ; — : — ; ■ . ■ =: sum required. ' 4 2.(n+\),{n+2) * 14. Required the sum (S) of the series ■^— h -r 1 ■-r» &c. continued ad infinitum, ► Let x=s — ^ and S:=s 2 ' 1+07 z Then --^j — ssX'^^x^+x^^-^x^+x'j tJ'r. i-jrx .4ndz^{\+x)X{x — x2+x3 — x^, eJ'c.) SUMMATION OF SERIES. 185 Whence^ by ^x — x^+x^'--~x^+x^^ ^c. :ey by C a icatioTiy ^ 1 ?7iultifilication^ C ^^"-^ X X^-\-X^'^X^'\-X^^ ^c. Whose sum is s= x+0+O+O+Oy ^c. Therefore z^=iX* X And JT— 072 + x^ — x^+x^^ ^c, = sum \+x' ,11111 41 2 4^8 16^32' 1+^ 3 required* 12 3 15. Required the sum of the series f- f- h 248 4 . 5 ^ + ^&C. 2 Let 07= -i-, awrf iS= 2 ' (1— ar)a' 7'^fn ri Y2=J^+2x2+3x3+4x4+5ar*, eJ'c. ^«(/ z=(l— .r)2x(a:+2a:2+3j;3+4^4+5x5^ ^Tc.) Whence^ i fnulti/iticati JVhence, by ^x+2x^+3x^+4x^f ^c. i/iticaUonj ^ 1 — 2j: + x* x+2xi+3x^+4,x^, Ufa, — 2x^ — 4x3 — Qx^^ l!fc, + x^+2x^,^c. Whose sum is ssx+O -fO + 0, ^c. There/ore x=sz, And J7+2x24-3x3+4x4+5x«, e^^c.! X (1— x)s* 12 3 4 5 4 2 ^ 4 ^ 8 ^ 16^32' n— 4)a required^ ^ 186 SUMMATION OF SERIES. 16. Required the 8um (S) of the infinite series , ere. 3^^ 9 ^27^81 Then '—^-—xax+4.x^+9x^+\6x^+2SxSy IsTc. ( 1— x)3 ^nrfz=.(l— a;)3x(r+4x2+9x3+16x^ ISfc.) ^=X'\-x^y as will be found by actual multijilication. There/ore x+x^xsz. ^nd conaeguently x+^^+9x^+ 1 6x^, i:fc,^—r — "xT ^ I . 4 . 9 . 16 ,^ 7X(l+4) 3 I 3^9^27^81* (1— •J)3 2 2 9Um required, 17. Required the sum (S) of the infinite seues a a+d a+2d a+Zd m mr mr^ mr^ 1 5' Let JT a ag ■■, owe/ 5 4 w.(l — ^)2 rrt^ f g , a+d ^ a+2d , a+3d ^^ m (1— x)2 m Twr mr^ mr^ ^ r . n+rf . a+2d , a+3d ,_ (» — ^r {ii+3rf)XJ^', ^c. ^nrf z=(l— a:)2x(a+Ca+r/)r+(a+2d>2+(a+3rf) x3),^c.s»(l-^)Xa+rfJ?>«« ««// afifiear by actual multi- filication. Therefore z^[\ — :p)Xo+c/x. SUMMATION OF SERIES. isr > , . a , d+d , a+2d . . jfnd consequent ly^ — }- — -| r-, cTc. = iw tnT iwr* (1 — ^x)Xfl+rfj: -:= «wm q/'Me injinite aeries required t EXAMPLES FOR PRACTICE. 1. To find the sum of n terms of the series a+(a-^^) -f(a— 2cf)+(a— 3(/)+(a— 4rf), &c. Jna, — X(2a— 7J — IXrf). 2. Required the sum of the infinite series a+rfa-f d^+d^a+d^a^ &c. where cf is a proper fraction. Ana. ■ 1— rf S. Required the sum of the infinite series l + 3x+ 6ar»+lOj;3+15x<, Etc. ^ I 4. Required the sum of the infinite series l+4x+ I0x3+20:c3-^35^4j &c. . 1 (I— ;c)4- 5. Required the sum of the infinite series r-r+ — + l«o 3*5 -+ ~, 8CC. ^«.. -. 6. Required the sum of 40 terras of the series (1X2) +(3X4)+(5X6)+(6x7)+(rx8), &c. Ana, 22960. 7. To find the sum of the infinite series 1+2^x4- 34^2+44x3+54 1:4, 8cc. ^' l + llj?+Ux>+ll.r3 Ana. , , ' ' \j — ■ ■. 8. Required the sum of the infinite series -7-r-rr ^ 2.3.4.5 S.4.5.6 ' ^»*. 188 SUMMATION OF SERIES. i 9. Required the sum of n terms of the series I 2.3.4. 5. 6„ -+ -+ T3+ 7J+ T5+ ^' ^'^' ^ r 7' ^n«. — r—l r(r— I) 10. Required the sum of n terms of the series I . 1 . 1 f :rTT^+ TTT^> &c. 1. 2.3.4 2.3,4.5 3.4.5.6 . J ^ '**' 18 3.(«+ . {71 + 2) . (»+ ^> 1 1 \ 11. Required the sum of the series— -+ —- + -^r; 1.4 A,5 —3.0 ^ 4J* ^""^ nX(3+w)- 11 . n , n TE. 18* 3+37i 12+6« 17+ ^n 12. Required the sum of the series —- ^ -f 2.6 4.8 1 c ^ ^ ^ 2 o .5n+3«* -, &c. r-7 — ■ . vfw«. 2= — , o=- 6.10' 2«(44-27i) 16' 32+48»4-iew2 1 1 13. Required the sum of the series h ^ 4.8 6.10 1 ^ 1 8.12 ' • (2-f-2w).(6+2«)* 48 16-f-16« 36 + 24« 14. Required the sum of the series 1 1* 3*8 6.12 I . 1 i 9.16 12.20' 3w.(4+4w) ^729. Sssr , iSsss— .. 12 12+12» SUMMATION OF SERIES. 189 15. * Required the sum of the infinite series 1 + — , &c. 3 4 5 6 jina, .69314718, ^c. or hyfi, log. of2» I " 16. Required the sum of the infinite series 1— -— + , &c. 5 7 9 11 jina. .78539, &c. or \ cir.ofthe circle whose diam, is 1. 1 2 17. Required the sum of the diverging series — 3 4 5 2 3 -I — — I , &c. w^«*. .193 147, ISTc, or l+hy/u log. of 4 5 6 a 1 18. Required the sum of the diverging series 22 32 42 52 5 Jns. 1.943147, 8cc. or 1- hyfi. log, of 2. 4 19. tRequired the sum of the hyper-geometrical series 1—14.2 — 6+24—120, 8cc. or 1 — 1a+2b — 3c + 4d — -5e, Sec. ina, near afifir ox. -value ss.298174. * A great variety of series, of different forms, may be found in other authors ; but those which are here given ^vUl be sulEcient for the learner's practice. The names of the principal authors, who have written upon this subject, are as follows : Archimedes ; Arabes ; D* Alembert ; Barrow ; Briggs ; Nicholas, Daniel, John, and James Bernoulli : Fermat; Descartes; Clairaut; C<mdorcet{ Cotes; Dodson; Euler; Emerson; Fagnanus; Le Grange; Goldbach ; Gregory; Halley; Harriot^ Huddens; Huy- gens; Button; Kepler; Keil ; Landen; Madaurin; DeLagney; Leibnitz ; Lorgna ; Lucas de Burgo ; Manfredi ; Monmort ; De Moivre t Montano ; Nichole ; Newton ; Oughtred ; Riccati ; Reg- nald; Saundersoni Sterling; Slusius; Simpson; Brook Taylor; Varignon; Vieta; WalUs; Waring; &c. t For an account of these series, with a new method of finding thcdr i^iproximate values, see Hutton^s Mathematical Tracts, lately published. [ 190 ] OF LOGARITHMS^. Logarithms are numbers so contrived and adapted tea other numbers, that the sums and differences of the former shall correspond to, and show, the products and quotients of the latter. Or, more generally, logarithms are the numerical exponents of ratios ; or a series of numbers in arithme- tical progression, answering to another series of numbers in geometrical progression. 7y 5^' ^' ^' ^' ^' ^' Indices or logarithms. ' ^ 1, 2, 4, 8, 16, 32, Geometric firogression. Or 5^' ^' ^' ^' ^' ^' Indices or logarithms, ' C 1> 3, 9, 27, 81, 243, Geometric progression. Or 5^' ^' ^» ^» ^' ^' Ind.orlog. ' C 1> 10, 100, 1000, 10000, 100000, Geo. prog. Where it is evident that the same indices serve= equally for any geometric series ; and consequently^ there may be an endless variety of systems of logarithms^ 10 the same common numbers, by only changing tht^ second term, 2, 3, or 10, &c. of the geometrical series * The invention of logarithms is the undoubted right of Lortf Napier, Baron of Merchiston, in Scotland, and is properly considered as one of the most useful and excellent discoveries of modem times. A table of these numbers was first published by him at Edinbuighf ann. 1614, in a treatise entitled Canon Mirificum Logarithmonakf and, as their great utility and extensive application were sufficiendy appa. rent, they were immediately received by all the learned throughout Europe, Mr. ffenry BriggSt Saviiian professor of Geometry at Chcfordf OF LOGARITHMS. 191 It is also apparent, from the nature of these series, ^t, if any two indices be added together, their sum '^l be the index of that number which is equal to the Pfoduct of the two terms, in the geometric progression, '0 which those indices belong. Thusy the indices 2 and 3, being added together^ are = ^ i and the numbers 4 and 8, or the terms corresponding ^ith those indices^ being multifilied together^ are = 32, ^/dch is the number answering to the index 5. And, in like manner, if any one index be subtracted K*om another, the difference will be the index of that ^ umber which is equal to the quotient of the two terms ^ which those indices belong* Thua^ the index 6, minus the index 4, {> e= 2 ; and the ierma corres/ionding to those indices are 64 and I6y whose quotient is =: 4> ; which is the number answering (o the index 2, upon hearing of the discovery, set out upon a visit to the noble in- ventor, and soon afterwards they jointly undertook the arduous task of computing new tables upon this subject, and reducing them to a more convenient form than that which was at first thought of. But lord Napier dying before they were Hnished, the whole burden fell upon Mr. Briggs^ who, with prodigious labour and great skill, made an entire Canon, according to the new form, for all numbers from 1 to 20000, and from 9000 to 101000, to 14 places of figures, and published it at London, in the year 1624, in a treatise entitled yfriV/;. metica Logarithmica, with directions for supplying the intermediate ckiiiads. This Canon was again published in Holland by Adrian Vlacq, aifho 1628, together with the logarithms of all tlie numbers which Mr. Bn'ggs has omitted ; but he continued them only to 10 places of decimals. Mr. Briggt also computed the logarithms of the sines, 'tangents, and secants to every degree, and _ i^ part of a degree of the whole quadrant, and subjoined them to the natural sines, tan. gents, and aecants, which he had befoie computed to 15 places of 192 OF LOGARITHMS. For the same reason, if the logarithm of any number be multiplied by the index of its power, the product will be equal to the logarithm of that power. Thusy the index or logarithm q/*4, in the above aerte»f ia 2 ; and^ if this number be multifilied by S, the firoduct wil 6r = 6; which ia the logarithm of 64j or the third flower of 4. And, if the logarithm of any number be divided by the index of its root, the quotient will be equal to the loga- rithm of that root. Thua, the index or logarithm of 64 ia 6 ; andj if thia number be divided by 2, the quotient will 6^ = 3; which ia the logarithm ofS^ or the aguare root 0/64, The logarithms most convenient for practice are such as are adapted to a geometric senes increasing figures. And these tables, together with their construction and use, were first published in the year 1633, after Mr. £riggs*s death, by Mr. Henry Gellibrand, under the title of TVigoiwmetria Britamuea. Benjamin Ui sinus has also given us a table of logarichms to every 10 seconds. And Mr. Wolf^ in his Mathematical Lexicon^ sayv that one Van Loser had computed them to every single second, but his untimely death prevented tneir publication. A great numbet of otlur authors have treated on this subject, but as their numbers arc freqaently inuccurate and incommodiously dis- posed, they are now ger>eral7y iieglecied. The tables in most repute at present axe those of Gardiner, in 4to. first published in the year 1742, and Sherwin's tables. In 8vc. first printed in the year 1705, where the logarithms of all numbers may be easily found from 1 to 1000000 ; and those of the sines, tangents, and secants, to any de- gree of accuracy required. Dodson's Jntilogarithmic Canon is likewise a veiy ingenious workt being of ^reat use for finding the numbers answering to any given logarithms. Smoe the first publication of this wodk, Mr. Miehad Taylor** tables have appNeared, coniainmg the common logarithms, and the loga- jSthmic unes and tangents to every second of the quadrant. OF LOGARITHMS. 193 in a tenfold proportion, as in the last of the above forms ; and are those which are to be found, at present, in most of the common tables upon this subject. The distinguishing mark of this system of logarithms is, that the index, or logarithm, of 1 is ; that of 10, 1 ; that of 100, 2 ; that of 1000, 3, &c. And in decimals the logarithm of 1 is — 1 ; that of .01, — 2 ; that of .001, — 3; &c. From Whence it follows that the logarithm of any num- ber between 1 and 10 must be and some fractional partSi and that of a number between 10 and 100 will be 1 and some fractional parts; and so on for any other number whatever. And since the integral part of a logarithm is always thus readily finind, it is usually called the index, or charac* teriatic ; and is commonly omitted in the tables ; lieing left to be supplied by the operator himself, as occasion re* quires. OF THE „ MAKING OF LOGARITHMS. Whatever arithmetical progression we apply to a geo* metrical one, the terms of it are logarithms only to that leries to which we apply them, and answer the end pro* posed only for those particular numbers ; so that if we had logarithms adapted only to particular geometrical seriesi they would be bat of little use. The great end and de- sign of these numbers is the ease and expedition which they afibrd in long calculatbns, by saving the laborious work of mtilii/Uicafioni diviaion^ and the extraction of r%ot% i but this end would never be completely answereli R 194 OF LOGARlTHxMS. unless logarithms could be adapted to the whole system of numberst 1)3,3, 4, &c. And as here lies the chief excel- lence and merit of the contrivance, so also the difficulty. For the natural system of numbers, ly 2, 3, 4, &c* being an arithmetical, and not a geometrical series, seems rather fit to be made logarithms of, than to have logarithms ap* plied to it. But this difTiculty may be easily removed, by considering. That though the whole system of natural numbers, I, 3, 3, 4, &c. is not in geometrical progression, and cannot, by any means, be made to agree with such a series, yet they may be brought so near it, as to be within any assign- able degree of approximation ; which may be conceivtrd, in general thus : supixise a fraction indefinitely small to be represented by J7,atid a geometrical series arising from 1, in the ratio of 1 to 1 +^> to be 1, (1 + x)*, (1 + jr)2, ( 1 + ^)^>( 1 + ^)^j &c. Then must some of these tenns come indefinitely near to all the natural numbers, 1, 2,3, 4, &c. ; because, amongst numbers which arise by ex-^ tremely small increments, some of them must exceed, or^ fall short, of any determinate number, by an mdefiniteljr^ little excess or defect.. If, therefore, in the places of the terms of this series^ which approach indefinitely near to any of the natural numbers, we suppose these natural numbers themselves to be substituted, then will this series be in geometrical progression, to an exactness which may be caUed^'ncf^ntf r; because the approximation of its terms to the natuni numbers can never end, but goes on in infiniium. And since this imagined geometric series-comprehendi^ xkidefinitely near, the whole system of natofalnilmblrrs; Iv 3*^ 3, 4) &c. so the indices of its terms comprehend a- Vhfile system of logarithmsi which arer adapted- to this OF LOGARITHMS. 195 system of numbers, and may be extended to any length we please. For though the natu^i system of numbers make not, by themselves, a complete j^eometrical series, yet they are conceived as a part of such a series, and their lopjarithms arc tlie indices of their distances from upity in that series ; or, more generally, they are the correspond- ing terms of an arithmetical series applied to that geome- trical one. But, again, it must be observed, that an indefinitely small fraction cannot he assigned ; and« therefore, in the actual construction of logarithms, we must be contented with a determinate degree of approximation. Whence, according as we take x^ in the series, 1, ( I + J^)S (1 +J^)*> (1 + jr)3, (1 + xy, &c. the approximation of its terms to the natural numbers will be in different degrees of exact- ness: for the lessor is^ the nearer will be the approxima- tion ; but then the more are the number of involutions of 1 + ^> necessary to come within any determinate degree of nearness to the natural number assigned. Thus then we uaay conceive the ix)S8ibility of making logarithms to all the natural numbers^ 1, 2, 3, 4, &.c. to any determinate degree of exactness; viz. by assigning a very : small fraction for jt, and actually raising a series, in the ratio of 1 to 1+ ^9 iind taking for the natural numbers such terms of that series as are nearest to them, and their indices for the logarithms. Hut then, to construct loga- riihms in this manner, to suqh an extent of numbers, and degree of exactness, as would be necessary to make them of any considerable use, is next to impossible, because of the almost infinite labour and time ii would require. This, however, is an introduction for understanding the method of the nolfle inventory who undoubtedly first took the hint 196 OF LOGARITHMS. of making logarithms from the consideration of the in- dices of a geometrical series; and by means of the prin- ciples and known properties of these progressions he first formed his tables, and adapted them to the practical pur- poses intended. PROBLEM L To find the logarithm of any of the natural numbers^ 1, 2, 3, 4, ^c, according to the method of Napier. ; RULE*. 1. Take the geometrical series, 1, 10, 100, 1000, 10000, Sec. and apply to it the arithmetical series 1, 2, 3, 4, &c. as logarithms. 2. Find a geometric mean between 1 and 10, 10 and 100, or any other two adjacent terms of the series betwixt which the number proposed lies. 3. Between the mean, thus found, and the nearest ex- treme, find another geometrical mean, in the same man- ner ; and so on, till you are arrived within the proposed li- mit of the number whose logarithm is sought. 4. Find as many arithmetical means, in the same order as you found the geometrical ones, and the last of these will be the logarithm answering to the number required. * The reader who wishes to inform himself more particularly concerning the history, nature, and construction of Ic^nthroi, may consult Hutton's Mathematical Tables, lately published, where he will find his curiosity amply gratified. OF LOGARITHMS. 197 EXAMPLES* Let it be required- to find the logarithm of 9. . Here the numbers between which 9 lies are 1 and 10. J^'irsty then, the log, af\0 ia L, and the log, qf\ t« 0; 1 + ther^ore - " w. ^» ss .5 i% the arithmetical mean, and y/{\ X IO)=v^lO=3.l622r77 = 5'co»jtfrricmea«; whence thv logarithm of S A 622 777 is .5. Secondly y the log. of \0 is 1, and the log, of 1 4. 5 3.1622777 is .5 ; therefore — =s .75 a= arithmetical jnea7iyand^{\0 X 3.1622777) = 5 6234132 =s geometric iHciin: whence the log. q/*5.6234132 is ,75. Thirdlyy the log, of \0 is 1, and the log, (/ 5.6234 132 1+75 fs .75-f t her (fore — ■ = .875 =as arithmetical mcan^ <and 5/(H).X5.623413:;i) s 7 A9S942\ss geometric mean : whence the log. (/ 7.498942 1 is i875. Fourthly, the log. of Wis l, and the log. of 7.498942 1 is .875 ; therefore — r-^ « .9 $7 S ^^ arithmetical mean, and ^( 10 X 7.4989421) SB 8.6596i43 1 «»/8r«ro»iff/nc mean: whence the log. qftj659^3\ is .9375. Fijthly, the log. of 10 i* I, and the log, of 8.6596431 1 -^ 93T5 is ;93r5 ; .thert^fort ^^^ le .96875 « arithmetical mean, and •(lO X 8.6596431) =9.3057204 »:5'Mm(?m'c mean: whence tht log. of 9-3057204 is .96875. 'R2^ / / c» 19S OF LOGARITHMS. Sixthly^ the log. 0/ 8.6596431 is .^STS, and the log, of .9375 4- 96875 9.3057204 w .96875 ; therefore ^ =.953125 =flWrA.mean,andv^(8.6596431x9.3057204)=8.9768713 ^=t geometric mean: whence the log, qf 8 .97687 13 is .953125. jind^tjiroc ceding in this manner^ after 25 extractions^ the logarithm o/" 8.9999998 Vfill be found to be .9542425 ; which may be taken for the logarithm of% because it dif fers from it only by 505Jofl6 > and is therefore sufficiently exact for all firacticatfiurfioses. jind in the same manner the logarithms of almost all the prime numbers were found i a work so incredibly laborious^ that the unremitted industry of several years was scarcely sufficient for its accomfilishment, PROBLEM IL To determine the hyfierbolic logarithm (l) (f any give?i number Csy, s - ■>. The hyperbolic logarithm pf any number fs)the^dex) of that'term of the logarithmic progression which agfees with the proposed number multiplied by the excess of the common ratio above unity. * Let, therefore, (I + x)« be that term of the logarith- mical progression, 1,(1 + x)\ ( 1 + x^^ ( 1 + x)», ( I + j:)S Sec. which is equal to the required number (n). Then wiU (I + jc)« = n, and 1 + a? = «»; and if I + y be put as N, and m =58 — ^ wc shall have 1 -f x = N« == (1 + y)»» 1 + my + m X^i^ya + m K^Ii^ OF LOGARITHMS- 199 m — 1 m — 2 And, consequently, x =fny+mx — —y^+^nX—^ — X y\ &c. where m being rejected in the factors Tn-I- 1, »2 — 2, m — 3, &c. being indefinitely small in com- parison of I, 2, 3, &c. the equation will become ;cs=7/iy — 2^3-4' Hence _ (n:r = i.)= y — 1- +' ^ — L. + ^ &c.= m 2 3 4 5 hyperbolic logarithm of n, as was required. PROBLEM IIL The hijfierbolic logarithm (l) of a number being gtven^ to Jind the number (n) itself^ which answers to it. Let (14-^)" be that term of the logarithmic progres- sion 1, (1 + ^)S (1 +^)S i^+^Yi (1 + J^)S &c. which is equal to the required number n. Then, because ( 1 + x)" is universally = 1 + war + n x ^ — i « . w — 1 w — 2 , ^ .... — r — x2+7zx — - — X — - — J^^i occ. we shall have I + nx n — 1 n — 1 n — 2 + nx -T--^^ + nx — T— X — T— -^^ ^c» = N. But since tz, from the nature of the logarithms, is here t supposed indefinitely great, it is evident that the numbers ^( ^' connected to it by the sine — may all be rejected, as fiir ^ as any assigned number of terms* For as 1, 3, 3, &c. are indefinitely small m compariflion to n, the rejecting of those numbers can iPcry litCle affect the values to which tliey betoog • » 200 OF LOGARITHMS. If; therefore, 1, 2, 3, Sec. be thrown out of the factors — r — I — - — , , &c. we shall have 1 + nx -J + "TTTT + ;: J &c.= N. 2.3 2.J.4 But nx (= l) is the hyperbolic logarithm of (1 + :r)«, or K, by what has been before specified ; and therelbre 1 1% l3 1.4 + L + — H + ~ — » &c.=s N = number required. 2 2,3 2<3.4 PROBLEM IV. To determine the hyfierboUc logarithm (l) of any j*iven number (n) by a universally convergiiig aeritu, t/2 y3 yA The series y — ~ + -^ , &c. is the most easy and natural that can be obtained ; but, in determininp^ the logarithms of large numbers, it is but of little use, since, in such cases, it diverges instead of converging. Let, therefore, the number whose logarithm you would find be denoted by , and also let (1 + •^)" be the i — y . term of the logarithmic progression agreeing with the proposed number. Then (I + x)n = -L-; or 1 + ^ = _L^x «, (I— y)-^ = (l— y)"*(by putting m== )a»l— my+ n w— I m — 1 m-— 2 „ ^ m X — T— y».-^m X — T— X — r— y3,»kc. OF LOGARITHMS. 201 Whence y + -- H 1 , &c. = = nxasz hy- ^2^34' m ^ perbolic logarithm of ; which series, it is manifest, \ — y will constantly converge, let the value of be ever so great ; because y will always be less than unity. But it is to be observed, that this series, except in its signs, has exactly the same form with that above found for the logarithm of 1 + y, and that, if both of them be added together, the series 3y + 9y3 2t/* 2r/^ . . — ^ H — '• — h -~-, Sec. thence arising, will be more simple than either of them^ as one half of the terms will, in that case, be eniirely destroyed. Since, therefore, the sum of the logarithms of any two numlws is equal 'to the logarithm of the pro- duct of those numbers, it is manifest that 2x -f 2^3 2j7* 1 -4- X —^ + ■— , Sec. will truly express the logarithm of ; which series converges still faster than jc -] 1 , Sec. not only because the even powers are here destroyed, but because x, in finding the logarithm of any given number (n), will have a less value. And, in order to determine what this value of x must \ -{- X N— — I be, make , = n, and then x will be found = — r-^: 1 X N 4" 1 p but if the quantity praposed — be a fraction, instead of a whole number, make =» — > 1 _j7 q 202 OF LOGARITHMS. P *— Q and we shall have 3 = i; and either of these values being substituted in the foregoing series 2jc + r- — I > &c. will give the hyperbolic logarithm of the number required. Now, by finding JVa/ner'a logarithm of any number, according to the foregoing method, Biigga^Si or the com- mon logarithm of the same number, may be found as follows : Briggs's logarithm of any number is to J\/a/iter*8 loga- rithm of the same number, as liriggs'a logarithm of 10 is to J\ra/tier*8 logarithm of 10. But Brigga*s logarithm of 10 is 1, and JVa/iier's loga- rithm of 10 is 2,302585093; whence, if Brigga's^or the common logarithm of any number, be denoted by c. l. and JVafiier^Sy or the hyperbolic logarithm of the same number, by h. l. we shall have 2.302585093 : 1 : : I H. L. : c. L,; or h. l. X = H. L. X 2.O02585093 .4342944819 =« c. l. as was required*. * There are, besides these, many other ingenious methods, which later writers have discovered, for tiiidmg the logaiithms of numbers in a much easier way than the original inventor; but, as they can- not be understood without a knowledge of some of the higher branches of the mathematics, 1 have thought proper to omit them, and must beg le ve to refer the reader to those works which are written expressly upon the subject. It would likewise much exceed the limits of this compendium, to point out all the peculiar artifices that are made use of for construct, ing an entire table of these numbers ; such as thofe of Gardimr, Skerviin^ and others, who have treated on this subject ; but any in- formation of this kind, which the learner may wish to obtain, may be found in JButton'a Tables, before mentioned. OF LOGARITHMS. A TABLE OF LOGARITHMS. Havinf^ explaineil the mcihotl of making a table of the logariifims of numbers greaiir than imity, ibc next thing to be done is lo show how the logantlims of frac- tional quantities may be found. And, in order lo Ihisi it may be observed, that, ai we hjve hitherto supposed a ^omclric series to increase from a unit on the right hand, to we may now suppose il to decrease from a unit towards the left; and (he indices, in this cascj being made nei;ittive> will still exhibit the logarithms of the terms to which ihcy belong. r/ms-T-of. — .1— 2 — 1 0+ I + a + 3, ts'r. ^"'"■TWTil tV ' '" 10O lOOO.iS'c. Whence -\- 1 ieihf t'.gaH'hm of\0,and — 1 the logarithm c/^j ; +11 lite logartl/im of IQO, and — 2 the losarilhm 4 And from hence it appears, that all numbers, consisting of the same figures, whether they l« integral, fracttonaU or mixed, will have the decimal parts of their logarithms will be sunicici fm ghid, X wof .1 foond only by mfxns of jddiiion and it 3t l lO-=t-B-)-i.. 5; L,3 = L 10 — t_3i L. 6i=L.3+i_ 3i u>d 10 on for uiy olhcr of tboe numlien. In like niuiHOr the Ion of a X A ^ i- <■ + I" ^1 <h* leg. of s-^A^L. a — L. A| th«log. of r" sBHi^r, and ilial ih« bg. of 204 OF LOGARITHMS. Thu9ythe logarithm of 587^ being 3.7689339, the loga- rithm of^ jjp T^cm ^^' fi^^^ ^f^^ ^^ *^ asfollofoa : Num. Logarithms* 5 8 7 4 5 8 7.4 5 8.7 4 5.8 7 4 .5874 .05874 .005874 3.7 6 8 9 3 3 9 2.7 6 8 9 3 3 9 1.7 6 8 9 3 3 9 07689339 — i.7 6 8 9 3 3 9 — 3.7 6 8 9 3 3 9 —3,7 6 8 9 3 3^9 From this it also appears, that the indea: or charac- teri9tic of any logarithm is always one less than the num- ber of figures which the natural number consists of; and this index is constantly to be placed on the left hand of the decimal p9rt of the logarithm. When there are integers in the given number, the in- dex is always affirmative ; but when there are no inte- gers, the index is negative, and is to be marked by a line drawn before it, like a negative quantity in algebra. Thus J a number having 1, 2, 3, 4, 5j ^c. integer filacesj The index of its log, is 0, 1,2, 3, 4, isfc. jlnd a fraction having a digit in thefilace of firimes^ ae* condsy thirds, fourths^ ^c. The index qfits logarithm will de •— 1, — 2, — 3| — 4, It may also be observed, that, though the indices of fractional quantities are negative, yet the decimal parts of their logarithms are always affirmative ; and all opera- tions are to be performed by them in the same manner'as by negative and affirmative quantities in algebra. \. OF LOGARITHMS. 205 In taking out of a table the logarithm of any number, not exceeding 100000, we have the decimal part by in- spection ; and if to this the proper characteristic be af- fixed, it will give the complete logarithm required. But if the number, whose logarithm is required, be above 100000, then find the logarithm of the two nearest numbers to it that can be found in the table, and say, as their difference : the difference of their 1 )garithm8 : : the difference of the nearest number and that whose logarithm is required : ibe difference of their logarithms, nearly} and this difference being added to, or subtracted from, the nearest logarithm, according as it is greatek* or less than the required one^ will give the logarithm required, fiearly. Thus, let it be required to find the logarithm of 367182*. The decimal fiart o/S67\ i«, by the tablcy 5647844 \and 0/3672 ia .5649027: .-. The 5 367100 is 5.5647844 > log. of I 367200 ia 5.56490-27 J Their diff. 100 0001 183 diff. JSTeareat JSTo, C 367200 Given .Vo. ^367183 18 diff. *■ This method, ^ing founded on the supposition that the loiga* rithm9 of all numbers between 367100 and 367200, increase or de. crease, equally, according to their distance from 367100 or 367200, is not strictly tfue, but nearly so; and the greater any numbers are, «ith respect to their difference, the nearer will those differences be proportional. And, therefore, though this method will not give the exact logarithm, yet it wlU be a very near approjutnation, aad is su^ciently exact for most practical puipop- b 206 OF LOGARITHMS. Therffore 100 : .0001183 : : 18 : .0000212. w^nrf 5.5649027— .0000212 = 5.5648815 = logarithm of ^67 \^% nearly . If the number consists both of integers and fractions, or is entirely fractional, find the decimal part of the loga- rithm as if all its figures were integral ; and this, being prefixed to the proper characteristic, will give the loga- rithm required. And if the given number is a proper fraction, subtract the logarithm of the denominator from the logarithm of the numerator, and the remainder will be the logarithm sought; which, being that of a decimal fraction, must always have a negative index. And, if it is a mixed number, reduce it to an improper fraction, and find the difference of the logarithms of the numerator and denominator, in the same manner as be- fore. In finding the number answering to any given loga- rithm, the index, if affirmative^ will always show how many integral places the required number consists of: and, if negative^ in what place of decimals the first, or significant figure, stands; so that, if the logarithm can be found in the table, the number answering to it will al- ways be had by inspection. But, if the logarithm cannot be exactly found in the table, find the next greater, and the next less, and then say. As the diff*erence of these two logarithms : the difference of the numbers answering to them : : the difference of the given logarithm and the nearest tabular logarithm : a fourth number; which added to, or sub- tracted from, the natural number answering to the near- est tabular logarithm, according as that logarithm is less or greater than the given one, will give the number re- quired, nearly. Thus, let it be required to find the natural number an- swer)' oganthm 5.5648815. OF LOGARITHMS, S07 The next less and greater logarithms^ in the table^ are 5.5647844 > The numbers C 367 100 5.5649027 3 answering ^367200 Th eir diff. .0001183 1 00 And 5.5649027 — 5.56488 15=.00002 12. Therefore .0001183 : 100 : : .0000212 : 18 nearly. IVhence 367200 — 18=377182= WMmd^r required^. MULTIPLICATION BY LOGARITHMS. BULK. Add the logarithms of the factors together, and their sum will be the logarithm of the product required. Observing to add what is to be carried from the de- cimal part of the logarithm to the sum of the affirmative indices : And that the difference between the affirmative and ne- gative indices is to be taken for the index to the logarithm of the product. * Directions, at large, for the using of logarithms, may be found in most of the common tables upon tliis subject. Shenoin'n Mathe- matical Tables, of the edition 1741 or 1742« are reckoned the most correct and convenient, for practk:al purposes, of any now extant, except those of Dr. Hutton, lately published; which, besides their accuracy, are much better arranged; and, in the two first degrees, the sines. &c. are given to every second He has also a table ot hy- perbolic logarithms, and several others equally usefol. 208 OF LOGARITHMS. EXAMPLES. 1. Let the number 256 be multiplied by 4. The log. 0/256 = 2.4082400 The log. of 4 = 0.6020600 Thefiroduct = 1024.... 3.0)03000 2. Let the number 8.5 be multiplied by 10. The log- o/S.5 =0.9294189 The log. of 10^ 1. 0000000 The firoduct = 85 .... 1 .9294 189 3. Let the number 46.75 be multiplied by .3275. The log. 0/46.75 = 1.6697816 The log. o/'.3275= — 1.5152113 The firoduct =± 15.31 ....1.1849925 4. Multiply 3.768, 2.053, and .007693 continually to- gether. The log. of 3.768 = 0.576 1 109 The log. qf 2.053 = 0.3 123889 The log. 0/ .007693 = — 3.8860997 Thefiroduct^ .059511... — 2.7745955 5. Multiply .5, .4, and .13 continually together. The log. of .5 =—1.6989700 The log. of .4 = — 1.6020600 The log. of.\2^ — 1.0791812 Thefiroduct =.024.. 2.3802112 OF LOGARITHMS. 2©9 DIVISION BY LOGARITHMS. RULE. From the logarithm of the dividend subtract the loga* rithm of the divisor, and the number agreeing to the re- mainder will be the quotient required. But observe to change the index of the divisor from ne- gative to affirmative, or from affirmative to negative, and tlren the diffisrence of the affirmative indices must be taken for the index to the logarithm of the quotient. And, also, when a unit is borrowed, in the left hand l)]ace of the decimal part of the logarithm, add it to the index of the divisor; but, if it be negative, subtract it: ai!d let the index ^^^^"8^ ^''O"^ thence be changed and worked with as before. EXAMPLES. 1. Let the number 56 be divided by the number ♦. The log. of 56 =s 1.7481880 The log. of 4 = 0.6020600 The quotient as 14 . . . 1.1461280 3. Let the number 50.75 be divided by the numbei* The log. of SO J 5 = 1.7054360 The log. of .25 = = — 1.3979400 The quotient b 203 ...#••.. 2.3074960 aid OP LOGARITHMS. 3. Let the number .24 be divided by the number SO. The log. of.24i^ — 1.380^1 12 nc log. ^ 80 a 1 .9050900 The quotient .003 .... — 3.4/71212 4. Let the number .01265 be divided by the number .35. The log. 0/ .01265 =« — 2.1020905 The log. of .35 sss — 1.7403627 The quotient = .023... 2.3617278 INVOLUTION BY LOGARITHMS. RULE*. 1. Seek the logarithm of the given number in the table. Jl 2* Multiply the logarithm, thus ^und, by the index of the proposed power. 3. Find the number corresponding to the product, and it will be the power required. JSTcte. In multiplying a logarithm with a negative in- dex, by any affirmative number, the product will always ' be negative: But what is to be carried &om the decimal part of the logarithm will always be affirmative: And therefore their difference will be the index of the product; and is constantly to be made of the same kinA with the greater. * The rule of proportion is performed by addmg the Ipgaiathiiv of the last-two terms, and subtncdng the h^garithm of the first: OF LOGARITHMS. 211 XXAMPLES. 4. Required the second power of the number 3.874. The log. of 3.874 a= 0,5881596 The index » 2 The fiovfer « 15.01 .... 1.1763192 , 2k Required the third power of the number 2.768. The log. of 2.768 = 0.442 1 66 1 The index = 3 The flower =a 5l,2l .. .. 1.3264983 3*. Required the third power of the number .79 1 6. The log. of .79 1 6 »— 1 .8985058 The index a 3 7'he flower ^ .4961 .. ..—1.6955174 4. Required the twel&h power of the number-1.539. The log. of 1.539—0.1872386 The index mB 12 The power » 176.6 .... 2.2468632 5. Required the 20th power of 1.05. wfnt. 2.6533, ISTc. 6. Required the 100th power of 1,05. Am. 131.50, e^r. 212 OF LOGARITHMS. EVOLUTION BY LOGARITHMS. RULE. k Seek the logarithm of the given number in th table. 2. Divide the logarithm, thus found, by the denomina- tor of the index of the root proposed. 5. Find the number corresponding to this quotient, and it will be the root required. Xute, — When the index of the logarithm, to be divid- ed, is negative, and does not exactly contain the divisor, increase it by such a number as will make it exactly divisible, and carry the units borrowed, as so many tens, lb the left-hand place of the decimal, and then divide as in whole numbers. EXAMPLES* 1 . Required the square root of the number 225. The log. of 225 = 2.352 1 825 Therefore 2)2.352 1 825 The root= 15 1.1760912 2. Required the square root of the number 1 50 1 , The log. of 1501 = 3.1763807 Therefore 2)3.1763807 The root = 38.74 1.588 1 903 MISCELLANEOUS QUESTIONS. 213 3. What is the cube root of the number .16631^5 ? The log. of . 1 66375 = — 1 .22 1088 1 T/iert/orr 3) — 1 .22 1088 1 T/ie root = ,55 .... — 1 .7403527 4. What is the square root of the number .08 1 62 ? The log. of .08162 = — 2.9117966 Therefore 2) — 2.9U7966 The root =s ,2857 ... — 1 .4558983 5. What is the twelfth root of the number 176.6 ? The log. of 176.6 = 2.2469907 Therefore 12)2.2469907 The root » 1.539 1872492 MISCELLANEOUS QUESTIONS. 1. A person being asked what o'clock it was, an- swered, that it was between 8 and 9, and that the hour and minute hands were exactly together ; what was the time? h. ^ ^, Jim. 8 : 43 : 38y\. 2. Divide the number 50 into two such parts, that \ of one part, added to ^ of the other, may make 40. .,ins. CO and 30. 3. What two numbers are those, whose difference is 12; and their squares equal to each other? jina. + 6 and — . 6. 4. There is a certain number, consisting of two places, which is equal to the difference of the Sf|uarc8 of its* di- gits ; and if 36 be added to it, the digits will be inverted ; quxre the number ? Jnti. 4a. 214 MISCELLANEOUS QUESTIONS. 5. Given x^ + ySacSl, and y^ ■{- x^ = 17 ; to find x and y, ^na x = 3 and y = 2. 6. Given p' — xy = 666, and x^ + jt-i/ = 406 ; to find X and y, jins. x = 7 ««£/ i/ = 9. 7. Given the sum of three numbtrs, in harmonical pro- portion, = 26, and tlieir continued product = 576 ; to find the numbers. jln^. 12, S^and 6. 8. What two numbers are those, whose diff'crtncc, sum, and product are to each oilier as the numbers 2, 3, and 5, respectively ? ^fis,2a?id 10. 9. *To find that number whose cube being subtracted from its square, shall leave the greatest remainder possi- ble. ^«*. |. 10. It is required to find the least 3 whole numbers, so that ^ of the first, -^ of the second, and j^ of the third, shall be all equal to each other, jina, 28(j, 294, and 300. 1 1 . Given zx^ + xz^ a» 290, and jc^ + 2^ = 64 1 ; to find X and z. ^na. x = 5, and z ss 2. 12. Given the sum of three numbers in continued geo- metrical progression = 39, and the sum of their squares = 819; to find the numbers. *^na. 3, 9, 27. 13. Required the least number of weights, and the weight of each, that will weigh from one pound to 29 hundredweight, jina. 1, 3,9,27, 81, 243,729,fl«</2187. 14. Required two numbers such, that their sums shall be equal both to their product and the difference of their squares. ^na. 2.618034 and 1.618034. 1 5. It is required to find the least 4 affirmative integers such, that the square of the greatest may be equal to the sum of the squares of the other three, ^/^na, 3, 4, 1 2, and 1 3 . * This is properly a question in fluxions, but it is answered alge* braically by Mr. Emerson, as well as several others of the same nature. MISCELLANEOUS QUESTIONS. 215 1 6. If money be lent, at three per cent. To those who choose lo borrow, In what time shall I be worth a pound, If I lend a crown to-morrow I .4718, 46.90036 yearsy allowing com/i, int^ 17. Required the two least nonquadrate numbers, x and t/, such, that x2 + 1/2 and x^ + 1/3 shall be both square numbers. Ana, x = 364, and y = 273. 18. There are three numbers in geometrical proportion such, that, if (he mean be subtracted from the sum of the two extremes, the remainder* muhiplied by the sum of the said two extremes will be 9^ ; but, if that remainder be multiplied by the sum of all the three numbers, the pro- duct will be 133 ; it is required to find the three numbers by a simple equation. jint. 4, 6, and 9. 1 9. To determine two numbers whose sum shall be a cube, but their product and quotients squares. jlna, 4 and 14, 100 and 25, 900 and 100. 20. Required that arithmetical progression whose num- ber of terms is 10, sum of the terms 185, and the sum of the cubes of the terms 104525. Jins. 5,8, n, 14, 17, 20, 23, 26,29, 32. 21. To divide a given number (n) into 4 such parts, that if any other number («) be added to the first part, de- ducted from the second, multiplied by the third, and the fourth part divided by it, the sum, difference, product, and quotient, shall be all equal to each other. yn nn n nh x n 22. Given ar^y + t/^^- = 512500, and x^y — y^xtss 5500 ; to find xand y, Jina. xa3B23 and yss20. 23. Given x + ?/ + z = 6,xy + xr + yz = 11, and xyz^6i to find JT, I/, and r. j4na» xas 3, y=l,fl72c/2=2. 216 MISCELLANEOUS QUESTIONS. 24. To find two numbers in the ratio of 5 to 7, M^hicli, btfing respectively divided by 9 and 13, shall leave 3 and 8 for remainders. Ana. 2\Qy and 294. ,25. To find three numbers such, that | the first, \ the second, and \ of the third, shall be = 62 ; l of the fii-stj-J of the second, and •} of the third «= 47 ; and \ of the first, \ of the second, and -J of the third = 38. Ana, 24, 60, a;2e/120. 26. A, B, and C, are to share 100,000 pounds between them, in the proportion of ^, ^J, and -J, respectively ; but C's part being lost by death, it is required to divide the whole sum properly between the other two. Ana. Wafiareia 57142||f/. and B's 42857/^/. 27. To find fdur numbers, z^yyZy and w, having the pro- duct of every three given; viz. Jcyz = 231, jryw = 420, yzw = 1540, and a:zw = 660. Ana, JT 3= 3, y = 7, z = 1 1, and w = 20. 28. To find four numbers in geometrical proportioD, whose sum is 15, and the sum of their squares 85. Ana. 1,2,4,8. 29. To find three numbers, x, y, and z, when the pro- ducts of each by the sum of the other two are given ; viz. ^ X (y + z) = 48, y X (x -f z) = 39, and z x (x + y) = 63. Ang. J? = 4, y = 3, and z = 9. 30. What number is that, which, being any how divid- ed, the square of one part, when added to the other part, $hall always be a square number ? Ana. 1 oniy. 31. Given y3^2--i27,y3-f j;=l35, andx^ + ys ^ z^ fml\33i to find ;r, y, and z. Ana. x=:\0yyssSfandzss:2. .'32. Given jc^-f jcy=108,y2+ yz=69, and z^+ xz^ MO ; to find x, y, and z. Ana. xbs9, y=3, and z^=^0. S3t To find two mean proportionals between any two given numbers a and b. Ana. ^a^b and ^b^a. ■/ MISCELLANEOUS QUESTIONS. 217 34. Given \r+yz=s384, y+xz=23r, and z+:cy=s 192 ; to find a:, y, z. Ana. x=:10, y=17, awrf z=22. 35. To find the least number, which, being divided by 6, 5, 4, 3, and 2, shall leave the remainders 5, 4^ 3^ 2, and 1, respectively. Ans. 59. 36. To find three numbers such, that the sum or di£- fi^rence of any two of them shall be square numbers. Ana, 1873432, 2399057, ancf 2286168. 37. To find two square numbers such, that their sum may be a square, and their difference a cube, and the side of the said square and cube equal to each other. 38. To determine the number of fifteens that can be jiiade out of a common pack of 52 cards. Ana. 17264. 39. Given the rates a and b of two ingredients, and (he rate c of the compound m, to find what portionaf x and y of each must be taken to compose the mixture. Ana. xsamx r, yssmX-^ — r 40. Given x2+j:y+i/»=1087, and x*+x^y^+y^:=^ 4577295 ; to find x and y» Ana, x=z2\ and yssl7. 41. Given x+y+z^7Sj jr«+ya+z»=2546, and xy — j:z— yz&a527 ; to find Xj y, and z. Ana, xm:4i\j yssz2Syandz^9, s .1 42. Given a:+ys=5 152, and (x — y^X^x — t^)Y=8192 ;: to find X and y. Ana. a:sl08, and yss44^ 43. To find three numbers such, that if to the square of each the product of the other two be added, the sums <«hall be squares. Ana, 73, 9, 329* 44. Let the number of cards in a pack (/;)• be distributed into any number of heaps (w), by laying OS many cards upon the bottom heap as arc sufficienlt T ^■■ 2 1 8 MISCELLANEOUS QUESTIONS. to make up its number (9) ; then, by having the number of cards remaining in the dealer's hand (r) and the num* ber of heaps (n) given, it is required to find the sum of all the bottom cards. Ana, (q+\)xn'\-(r'^)s=i autnreqtured. V 45. To find 3 numbers such, that if each be subtracted from the cube of their sum, the remainders shall be cubes. Ana. fJlf^, ifffi, and ^^fj. 46. Given the cycle of the sun 18, the golden number 8, and the Roman indiction 10 ; to find the year. Ana* 1717. 47. To find 3 cube numbers such, that their sum shall be both a square and a cube number : and if that aitfn be squared it shall be a cube, and if it be cubed it jfcdl be a squai-e. 4^^ ^ t, ^ * ^25 . 8 '37^'216 48. To find 3 numbers such, that if each be added to the cube of their sum, their sums shall be cubes. •^"*- TTTjiTT* rrnn> ttttst* 49. With guineas and moidores, the fewest, which way Three hundred and fifty-one pounds can I pay? If paid every way 'twill admit of, what sum Do the pieces amount to ?— my fortune's to come. jina. 9 guineaa^ and 253 moidorea; and 37 waysf which ia ss 12987/. 1 . 2 50. Given x2/2*^=:z*^=slOO; to find the values of :f and y. Ana, x=:47.706 and zssslAi* 51. Given x^'— 21j?2*+147jc*=316, to find the va- lue ofx. Ana. :rd r' - ■■'■ «* MISCELLANEOUS QUESTIONS. 219 52. Given 44000xs+ 1 ss z^ ; to find x and z ia whole numbers. jins. xss:^04S29Sl22l7S\y and z^S49l78\\A%00\. 53« To find three whole numbers suchi that the ex- cess of the greatest above the middle number shall be to the excess of the middle number above the leasts as 3 to I : and also that the sum of every two of these shall be squares. jina. 4nX 1362, 4nX402, and 4nx82. 54. Given x+y^a (2), and j:«+y«=sA (32), to find x and y by quadratics. jins. aal.4697175, ancTy^.S 302824« 55. Given xy=500, and y*=s300; to find x and y. ^n«.xsa4.6914, and y=s5,Sl02. 56. Given xyx(j:+z)*=300,aprx(y+z)a= 1296, and t/rx(jr+y)««432 ; to find a:, y, and z. wfn«. jrss], y=3| z^9. 57. Given ii;3+j:+y+z = 57, w+j73+y+2-ai763, iif+x+y^+z^\:i50y and 5b;4- J^4-!/4-z3-b153; to find '^) Vi ^j and ti^. ^TM. XKs 14| y=s 1 1| zss5y and w^S. 58. Given jr + y=« 1750, J7z+yvs= 22708, xv+yz^^ 12292, and xzv+vzysx 159252; to find x, y, z, and v. ^na* x:b1743, y=s7| z^lS, and vmt7. 59. Given 5x+7y+9z»93256; to find all the difie- rent solutions in affirmative integers which the equation will admit of. jina. 13801148. 60. To find a square number such, that the sum of all its aliquot parts shall be a square number. Jina. 9401. 61. To find two sc^uare numbers such, that either of them, when added to its aliquot parts, shall make the same sum. ^na. 106276 and 16S6A9. 220 MISCELLANEOUS QUESTIONS. 63. To find 4 whole numbers such, that the difference 4)i every two shall be a square number. * ^72*. 1873432, 2288168, 2399057, and 6560657. 63. To find three numbers such, that if their sum be multiplied by the first, it shall be a triangular number, by the second a square, and by the third a cube. 64. To find three biquadrate numbers, the sum of which shall be a square. >4n8, 12^, 15S and 20^^. 65. To find a right-angled triangle such) that its peri- meter sliall be a cube, and tlie perimeter, together with the area, a square. jfna. Per]i.^^i^\ *Wf=V^'» >iy/'-=\VA'. 66. To find two different isosceles triangles su«h, that f-h^ir areas and perimeters shall be both equal. jitifi. Sides of the 072c=29, 29, 40. Ditto of the orAer=37, 37, 24, I* 67. There is an island 73' miles in circumference, and lliree fpotmen all start together to travel the same way about it: A goes 5 miles a day, B 8, and C 10: when \rfll they all come together again ? ^ns, 73 days^ 68. How much foreign brandy at 6s. fier gallon, and British spirits at 3s. fier gallon, must be mixed together, so that, in selling the compound at 9s. fier gallon, tlTb distiller may clear SO fier cent,? Ans, 51 gallons of brandy y and 14 of s/iii-iCfi^ THE END. "i -A^ i^l^^^^^BH THB NBW YORK PUBLIC LIBRARY 1 REFERENCE DEPARTMENT H Thii book i> tak ea from the BuUdIa« ■ r-mw ^^H k B IBj